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Theorem wemapwe 8538
Description: Construct lexicographic order on a function space based on a reverse well-ordering of the indexes and a well-ordering of the values. (Contributed by Mario Carneiro, 29-May-2015.) (Revised by AV, 3-Jul-2019.)
Hypotheses
Ref Expression
wemapwe.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
wemapwe.u 𝑈 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}
wemapwe.2 (𝜑𝑅 We 𝐴)
wemapwe.3 (𝜑𝑆 We 𝐵)
wemapwe.4 (𝜑𝐵 ≠ ∅)
wemapwe.5 𝐹 = OrdIso(𝑅, 𝐴)
wemapwe.6 𝐺 = OrdIso(𝑆, 𝐵)
wemapwe.7 𝑍 = (𝐺‘∅)
Assertion
Ref Expression
wemapwe (𝜑𝑇 We 𝑈)
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦   𝑤,𝐹,𝑥,𝑦,𝑧   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦   𝑤,𝑅,𝑧   𝑧,𝑆   𝑥,𝑈,𝑦   𝑥,𝑍
Allowed substitution hints:   𝜑(𝑧,𝑤)   𝐵(𝑧,𝑤)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦,𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)   𝑈(𝑧,𝑤)   𝐺(𝑧,𝑤)   𝑍(𝑦,𝑧,𝑤)

Proof of Theorem wemapwe
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wemapwe.u . . . . . . . . 9 𝑈 = {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍}
2 eqid 2621 . . . . . . . . 9 {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)} = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)}
3 eqid 2621 . . . . . . . . 9 (𝐺𝑍) = (𝐺𝑍)
4 simprr 795 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐴 ∈ V)
5 wemapwe.2 . . . . . . . . . . . 12 (𝜑𝑅 We 𝐴)
65adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑅 We 𝐴)
7 wemapwe.5 . . . . . . . . . . . 12 𝐹 = OrdIso(𝑅, 𝐴)
87oiiso 8386 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
94, 6, 8syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
10 isof1o 6527 . . . . . . . . . 10 (𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴) → 𝐹:dom 𝐹1-1-onto𝐴)
119, 10syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐹:dom 𝐹1-1-onto𝐴)
12 simprl 793 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐵 ∈ V)
13 wemapwe.3 . . . . . . . . . . . 12 (𝜑𝑆 We 𝐵)
1413adantr 481 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑆 We 𝐵)
15 wemapwe.6 . . . . . . . . . . . 12 𝐺 = OrdIso(𝑆, 𝐵)
1615oiiso 8386 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ 𝑆 We 𝐵) → 𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵))
1712, 14, 16syl2anc 692 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵))
18 isof1o 6527 . . . . . . . . . 10 (𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵) → 𝐺:dom 𝐺1-1-onto𝐵)
19 f1ocnv 6106 . . . . . . . . . 10 (𝐺:dom 𝐺1-1-onto𝐵𝐺:𝐵1-1-onto→dom 𝐺)
2017, 18, 193syl 18 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐺:𝐵1-1-onto→dom 𝐺)
217oiexg 8384 . . . . . . . . . . 11 (𝐴 ∈ V → 𝐹 ∈ V)
2221ad2antll 764 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐹 ∈ V)
23 dmexg 7044 . . . . . . . . . 10 (𝐹 ∈ V → dom 𝐹 ∈ V)
2422, 23syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐹 ∈ V)
2515oiexg 8384 . . . . . . . . . . 11 (𝐵 ∈ V → 𝐺 ∈ V)
2625ad2antrl 763 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐺 ∈ V)
27 dmexg 7044 . . . . . . . . . 10 (𝐺 ∈ V → dom 𝐺 ∈ V)
2826, 27syl 17 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐺 ∈ V)
29 wemapwe.7 . . . . . . . . . 10 𝑍 = (𝐺‘∅)
3017, 18syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐺:dom 𝐺1-1-onto𝐵)
31 f1ofo 6101 . . . . . . . . . . . . . . 15 (𝐺:dom 𝐺1-1-onto𝐵𝐺:dom 𝐺onto𝐵)
32 forn 6075 . . . . . . . . . . . . . . 15 (𝐺:dom 𝐺onto𝐵 → ran 𝐺 = 𝐵)
3330, 31, 323syl 18 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ran 𝐺 = 𝐵)
34 wemapwe.4 . . . . . . . . . . . . . . 15 (𝜑𝐵 ≠ ∅)
3534adantr 481 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐵 ≠ ∅)
3633, 35eqnetrd 2857 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ran 𝐺 ≠ ∅)
37 dm0rn0 5302 . . . . . . . . . . . . . 14 (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
3837necon3bii 2842 . . . . . . . . . . . . 13 (dom 𝐺 ≠ ∅ ↔ ran 𝐺 ≠ ∅)
3936, 38sylibr 224 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐺 ≠ ∅)
4015oicl 8378 . . . . . . . . . . . . 13 Ord dom 𝐺
41 ord0eln0 5738 . . . . . . . . . . . . 13 (Ord dom 𝐺 → (∅ ∈ dom 𝐺 ↔ dom 𝐺 ≠ ∅))
4240, 41ax-mp 5 . . . . . . . . . . . 12 (∅ ∈ dom 𝐺 ↔ dom 𝐺 ≠ ∅)
4339, 42sylibr 224 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ∅ ∈ dom 𝐺)
4415oif 8379 . . . . . . . . . . . 12 𝐺:dom 𝐺𝐵
4544ffvelrni 6314 . . . . . . . . . . 11 (∅ ∈ dom 𝐺 → (𝐺‘∅) ∈ 𝐵)
4643, 45syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝐺‘∅) ∈ 𝐵)
4729, 46syl5eqel 2702 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑍𝐵)
481, 2, 3, 11, 20, 4, 12, 24, 28, 47mapfien 8257 . . . . . . . 8 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→{𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)})
49 eqid 2621 . . . . . . . . . . 11 {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp ∅}
5015oion 8385 . . . . . . . . . . . 12 (𝐵 ∈ V → dom 𝐺 ∈ On)
5150ad2antrl 763 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐺 ∈ On)
527oion 8385 . . . . . . . . . . . 12 (𝐴 ∈ V → dom 𝐹 ∈ On)
5352ad2antll 764 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐹 ∈ On)
5449, 51, 53cantnfdm 8505 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom (dom 𝐺 CNF dom 𝐹) = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp ∅})
5529fveq2i 6151 . . . . . . . . . . . . 13 (𝐺𝑍) = (𝐺‘(𝐺‘∅))
56 f1ocnvfv1 6486 . . . . . . . . . . . . . 14 ((𝐺:dom 𝐺1-1-onto𝐵 ∧ ∅ ∈ dom 𝐺) → (𝐺‘(𝐺‘∅)) = ∅)
5730, 43, 56syl2anc 692 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝐺‘(𝐺‘∅)) = ∅)
5855, 57syl5eq 2667 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝐺𝑍) = ∅)
5958breq2d 4625 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑥 finSupp (𝐺𝑍) ↔ 𝑥 finSupp ∅))
6059rabbidv 3177 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)} = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp ∅})
6154, 60eqtr4d 2658 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom (dom 𝐺 CNF dom 𝐹) = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)})
62 f1oeq3 6086 . . . . . . . . 9 (dom (dom 𝐺 CNF dom 𝐹) = {𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)} → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→dom (dom 𝐺 CNF dom 𝐹) ↔ (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→{𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)}))
6361, 62syl 17 . . . . . . . 8 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→dom (dom 𝐺 CNF dom 𝐹) ↔ (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→{𝑥 ∈ (dom 𝐺𝑚 dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)}))
6448, 63mpbird 247 . . . . . . 7 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→dom (dom 𝐺 CNF dom 𝐹))
65 eqid 2621 . . . . . . . . 9 dom (dom 𝐺 CNF dom 𝐹) = dom (dom 𝐺 CNF dom 𝐹)
66 eqid 2621 . . . . . . . . 9 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))}
6765, 51, 53, 66oemapwe 8535 . . . . . . . 8 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ({⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} We dom (dom 𝐺 CNF dom 𝐹) ∧ dom OrdIso({⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))}, dom (dom 𝐺 CNF dom 𝐹)) = (dom 𝐺𝑜 dom 𝐹)))
6867simpld 475 . . . . . . 7 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} We dom (dom 𝐺 CNF dom 𝐹))
69 eqid 2621 . . . . . . . 8 {⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)}
7069f1owe 6557 . . . . . . 7 ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→dom (dom 𝐺 CNF dom 𝐹) → ({⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} We dom (dom 𝐺 CNF dom 𝐹) → {⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} We 𝑈))
7164, 68, 70sylc 65 . . . . . 6 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → {⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} We 𝑈)
72 weinxp 5147 . . . . . 6 ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} We 𝑈 ↔ ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) We 𝑈)
7371, 72sylib 208 . . . . 5 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) We 𝑈)
7411adantr 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐹:dom 𝐹1-1-onto𝐴)
75 f1ofn 6095 . . . . . . . . . . . 12 (𝐹:dom 𝐹1-1-onto𝐴𝐹 Fn dom 𝐹)
76 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑐) → (𝑥𝑧) = (𝑥‘(𝐹𝑐)))
77 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑐) → (𝑦𝑧) = (𝑦‘(𝐹𝑐)))
7876, 77breq12d 4626 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑐) → ((𝑥𝑧)𝑆(𝑦𝑧) ↔ (𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐))))
79 breq1 4616 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑐) → (𝑧𝑅𝑤 ↔ (𝐹𝑐)𝑅𝑤))
8079imbi1d 331 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑐) → ((𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))))
8180ralbidv 2980 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑐) → (∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))))
8278, 81anbi12d 746 . . . . . . . . . . . . 13 (𝑧 = (𝐹𝑐) → (((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
8382rexrn 6317 . . . . . . . . . . . 12 (𝐹 Fn dom 𝐹 → (∃𝑧 ∈ ran 𝐹((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑐 ∈ dom 𝐹((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
8474, 75, 833syl 18 . . . . . . . . . . 11 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∃𝑧 ∈ ran 𝐹((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑐 ∈ dom 𝐹((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
85 f1ofo 6101 . . . . . . . . . . . . 13 (𝐹:dom 𝐹1-1-onto𝐴𝐹:dom 𝐹onto𝐴)
86 forn 6075 . . . . . . . . . . . . 13 (𝐹:dom 𝐹onto𝐴 → ran 𝐹 = 𝐴)
8774, 85, 863syl 18 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → ran 𝐹 = 𝐴)
8887rexeqdv 3134 . . . . . . . . . . 11 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∃𝑧 ∈ ran 𝐹((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
8926adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐺 ∈ V)
90 cnvexg 7059 . . . . . . . . . . . . . . 15 (𝐺 ∈ V → 𝐺 ∈ V)
9189, 90syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐺 ∈ V)
92 vex 3189 . . . . . . . . . . . . . . 15 𝑥 ∈ V
9322adantr 481 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐹 ∈ V)
94 coexg 7064 . . . . . . . . . . . . . . 15 ((𝑥 ∈ V ∧ 𝐹 ∈ V) → (𝑥𝐹) ∈ V)
9592, 93, 94sylancr 694 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝑥𝐹) ∈ V)
96 coexg 7064 . . . . . . . . . . . . . 14 ((𝐺 ∈ V ∧ (𝑥𝐹) ∈ V) → (𝐺 ∘ (𝑥𝐹)) ∈ V)
9791, 95, 96syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝐺 ∘ (𝑥𝐹)) ∈ V)
98 vex 3189 . . . . . . . . . . . . . . 15 𝑦 ∈ V
99 coexg 7064 . . . . . . . . . . . . . . 15 ((𝑦 ∈ V ∧ 𝐹 ∈ V) → (𝑦𝐹) ∈ V)
10098, 93, 99sylancr 694 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝑦𝐹) ∈ V)
101 coexg 7064 . . . . . . . . . . . . . 14 ((𝐺 ∈ V ∧ (𝑦𝐹) ∈ V) → (𝐺 ∘ (𝑦𝐹)) ∈ V)
10291, 100, 101syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝐺 ∘ (𝑦𝐹)) ∈ V)
103 fveq1 6147 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝐺 ∘ (𝑥𝐹)) → (𝑎𝑐) = ((𝐺 ∘ (𝑥𝐹))‘𝑐))
104 fveq1 6147 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝐺 ∘ (𝑦𝐹)) → (𝑏𝑐) = ((𝐺 ∘ (𝑦𝐹))‘𝑐))
105 eleq12 2688 . . . . . . . . . . . . . . . . 17 (((𝑎𝑐) = ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∧ (𝑏𝑐) = ((𝐺 ∘ (𝑦𝐹))‘𝑐)) → ((𝑎𝑐) ∈ (𝑏𝑐) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐)))
106103, 104, 105syl2an 494 . . . . . . . . . . . . . . . 16 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → ((𝑎𝑐) ∈ (𝑏𝑐) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐)))
107 fveq1 6147 . . . . . . . . . . . . . . . . . . 19 (𝑎 = (𝐺 ∘ (𝑥𝐹)) → (𝑎𝑑) = ((𝐺 ∘ (𝑥𝐹))‘𝑑))
108 fveq1 6147 . . . . . . . . . . . . . . . . . . 19 (𝑏 = (𝐺 ∘ (𝑦𝐹)) → (𝑏𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑))
109107, 108eqeqan12d 2637 . . . . . . . . . . . . . . . . . 18 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → ((𝑎𝑑) = (𝑏𝑑) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))
110109imbi2d 330 . . . . . . . . . . . . . . . . 17 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → ((𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)) ↔ (𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑))))
111110ralbidv 2980 . . . . . . . . . . . . . . . 16 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → (∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)) ↔ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑))))
112106, 111anbi12d 746 . . . . . . . . . . . . . . 15 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → (((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑))) ↔ (((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
113112rexbidv 3045 . . . . . . . . . . . . . 14 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → (∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑))) ↔ ∃𝑐 ∈ dom 𝐹(((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
114113, 66brabga 4949 . . . . . . . . . . . . 13 (((𝐺 ∘ (𝑥𝐹)) ∈ V ∧ (𝐺 ∘ (𝑦𝐹)) ∈ V) → ((𝐺 ∘ (𝑥𝐹)){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} (𝐺 ∘ (𝑦𝐹)) ↔ ∃𝑐 ∈ dom 𝐹(((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
11597, 102, 114syl2anc 692 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → ((𝐺 ∘ (𝑥𝐹)){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} (𝐺 ∘ (𝑦𝐹)) ↔ ∃𝑐 ∈ dom 𝐹(((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
116 simprl 793 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑥𝑈)
117 coeq1 5239 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑥 → (𝑓𝐹) = (𝑥𝐹))
118117coeq2d 5244 . . . . . . . . . . . . . . 15 (𝑓 = 𝑥 → (𝐺 ∘ (𝑓𝐹)) = (𝐺 ∘ (𝑥𝐹)))
119 eqid 2621 . . . . . . . . . . . . . . 15 (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))) = (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))
120118, 119fvmptg 6237 . . . . . . . . . . . . . 14 ((𝑥𝑈 ∧ (𝐺 ∘ (𝑥𝐹)) ∈ V) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥) = (𝐺 ∘ (𝑥𝐹)))
121116, 97, 120syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥) = (𝐺 ∘ (𝑥𝐹)))
122 simprr 795 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑦𝑈)
123 coeq1 5239 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑦 → (𝑓𝐹) = (𝑦𝐹))
124123coeq2d 5244 . . . . . . . . . . . . . . 15 (𝑓 = 𝑦 → (𝐺 ∘ (𝑓𝐹)) = (𝐺 ∘ (𝑦𝐹)))
125124, 119fvmptg 6237 . . . . . . . . . . . . . 14 ((𝑦𝑈 ∧ (𝐺 ∘ (𝑦𝐹)) ∈ V) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) = (𝐺 ∘ (𝑦𝐹)))
126122, 102, 125syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) = (𝐺 ∘ (𝑦𝐹)))
127121, 126breq12d 4626 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) ↔ (𝐺 ∘ (𝑥𝐹)){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} (𝐺 ∘ (𝑦𝐹))))
12817ad2antrr 761 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵))
129 isocnv 6534 . . . . . . . . . . . . . . . . . 18 (𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵) → 𝐺 Isom 𝑆, E (𝐵, dom 𝐺))
130128, 129syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝐺 Isom 𝑆, E (𝐵, dom 𝐺))
131 ssrab2 3666 . . . . . . . . . . . . . . . . . . . . 21 {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍} ⊆ (𝐵𝑚 𝐴)
1321, 131eqsstri 3614 . . . . . . . . . . . . . . . . . . . 20 𝑈 ⊆ (𝐵𝑚 𝐴)
133132, 116sseldi 3581 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑥 ∈ (𝐵𝑚 𝐴))
134 elmapi 7823 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐵𝑚 𝐴) → 𝑥:𝐴𝐵)
135133, 134syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑥:𝐴𝐵)
1367oif 8379 . . . . . . . . . . . . . . . . . . 19 𝐹:dom 𝐹𝐴
137136ffvelrni 6314 . . . . . . . . . . . . . . . . . 18 (𝑐 ∈ dom 𝐹 → (𝐹𝑐) ∈ 𝐴)
138 ffvelrn 6313 . . . . . . . . . . . . . . . . . 18 ((𝑥:𝐴𝐵 ∧ (𝐹𝑐) ∈ 𝐴) → (𝑥‘(𝐹𝑐)) ∈ 𝐵)
139135, 137, 138syl2an 494 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑥‘(𝐹𝑐)) ∈ 𝐵)
140132, 122sseldi 3581 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑦 ∈ (𝐵𝑚 𝐴))
141 elmapi 7823 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝐵𝑚 𝐴) → 𝑦:𝐴𝐵)
142140, 141syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑦:𝐴𝐵)
143 ffvelrn 6313 . . . . . . . . . . . . . . . . . 18 ((𝑦:𝐴𝐵 ∧ (𝐹𝑐) ∈ 𝐴) → (𝑦‘(𝐹𝑐)) ∈ 𝐵)
144142, 137, 143syl2an 494 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑦‘(𝐹𝑐)) ∈ 𝐵)
145 isorel 6530 . . . . . . . . . . . . . . . . 17 ((𝐺 Isom 𝑆, E (𝐵, dom 𝐺) ∧ ((𝑥‘(𝐹𝑐)) ∈ 𝐵 ∧ (𝑦‘(𝐹𝑐)) ∈ 𝐵)) → ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) E (𝐺‘(𝑦‘(𝐹𝑐)))))
146130, 139, 144, 145syl12anc 1321 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) E (𝐺‘(𝑦‘(𝐹𝑐)))))
147 fvex 6158 . . . . . . . . . . . . . . . . 17 (𝐺‘(𝑦‘(𝐹𝑐))) ∈ V
148147epelc 4987 . . . . . . . . . . . . . . . 16 ((𝐺‘(𝑥‘(𝐹𝑐))) E (𝐺‘(𝑦‘(𝐹𝑐))) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) ∈ (𝐺‘(𝑦‘(𝐹𝑐))))
149146, 148syl6bb 276 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) ∈ (𝐺‘(𝑦‘(𝐹𝑐)))))
150135adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝑥:𝐴𝐵)
151 fco 6015 . . . . . . . . . . . . . . . . . . 19 ((𝑥:𝐴𝐵𝐹:dom 𝐹𝐴) → (𝑥𝐹):dom 𝐹𝐵)
152150, 136, 151sylancl 693 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑥𝐹):dom 𝐹𝐵)
153 fvco3 6232 . . . . . . . . . . . . . . . . . 18 (((𝑥𝐹):dom 𝐹𝐵𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑐) = (𝐺‘((𝑥𝐹)‘𝑐)))
154152, 153sylancom 700 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑐) = (𝐺‘((𝑥𝐹)‘𝑐)))
155 simpr 477 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝑐 ∈ dom 𝐹)
156 fvco3 6232 . . . . . . . . . . . . . . . . . . 19 ((𝐹:dom 𝐹𝐴𝑐 ∈ dom 𝐹) → ((𝑥𝐹)‘𝑐) = (𝑥‘(𝐹𝑐)))
157136, 155, 156sylancr 694 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥𝐹)‘𝑐) = (𝑥‘(𝐹𝑐)))
158157fveq2d 6152 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝐺‘((𝑥𝐹)‘𝑐)) = (𝐺‘(𝑥‘(𝐹𝑐))))
159154, 158eqtrd 2655 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑐) = (𝐺‘(𝑥‘(𝐹𝑐))))
160142adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝑦:𝐴𝐵)
161 fco 6015 . . . . . . . . . . . . . . . . . . 19 ((𝑦:𝐴𝐵𝐹:dom 𝐹𝐴) → (𝑦𝐹):dom 𝐹𝐵)
162160, 136, 161sylancl 693 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑦𝐹):dom 𝐹𝐵)
163 fvco3 6232 . . . . . . . . . . . . . . . . . 18 (((𝑦𝐹):dom 𝐹𝐵𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑐) = (𝐺‘((𝑦𝐹)‘𝑐)))
164162, 163sylancom 700 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑐) = (𝐺‘((𝑦𝐹)‘𝑐)))
165 fvco3 6232 . . . . . . . . . . . . . . . . . . 19 ((𝐹:dom 𝐹𝐴𝑐 ∈ dom 𝐹) → ((𝑦𝐹)‘𝑐) = (𝑦‘(𝐹𝑐)))
166136, 155, 165sylancr 694 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑦𝐹)‘𝑐) = (𝑦‘(𝐹𝑐)))
167166fveq2d 6152 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝐺‘((𝑦𝐹)‘𝑐)) = (𝐺‘(𝑦‘(𝐹𝑐))))
168164, 167eqtrd 2655 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑐) = (𝐺‘(𝑦‘(𝐹𝑐))))
169159, 168eleq12d 2692 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) ∈ (𝐺‘(𝑦‘(𝐹𝑐)))))
170149, 169bitr4d 271 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐)))
17187raleqdv 3133 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∀𝑤 ∈ ran 𝐹((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))))
172 breq2 4617 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐹𝑑) → ((𝐹𝑐)𝑅𝑤 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
173 fveq2 6148 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐹𝑑) → (𝑥𝑤) = (𝑥‘(𝐹𝑑)))
174 fveq2 6148 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐹𝑑) → (𝑦𝑤) = (𝑦‘(𝐹𝑑)))
175173, 174eqeq12d 2636 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐹𝑑) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑))))
176172, 175imbi12d 334 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝐹𝑑) → (((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
177176ralrn 6318 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn dom 𝐹 → (∀𝑤 ∈ ran 𝐹((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
17874, 75, 1773syl 18 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∀𝑤 ∈ ran 𝐹((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
179171, 178bitr3d 270 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
180179adantr 481 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
181 epel 4988 . . . . . . . . . . . . . . . . . . 19 (𝑐 E 𝑑𝑐𝑑)
1829ad2antrr 761 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
183 isorel 6530 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑐 E 𝑑 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
184182, 183sylancom 700 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑐 E 𝑑 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
185181, 184syl5bbr 274 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑐𝑑 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
186152adantrr 752 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑥𝐹):dom 𝐹𝐵)
187 simprr 795 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝑑 ∈ dom 𝐹)
188 fvco3 6232 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥𝐹):dom 𝐹𝐵𝑑 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = (𝐺‘((𝑥𝐹)‘𝑑)))
189186, 187, 188syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = (𝐺‘((𝑥𝐹)‘𝑑)))
190162adantrr 752 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑦𝐹):dom 𝐹𝐵)
191 fvco3 6232 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦𝐹):dom 𝐹𝐵𝑑 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑑) = (𝐺‘((𝑦𝐹)‘𝑑)))
192190, 187, 191syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝐺 ∘ (𝑦𝐹))‘𝑑) = (𝐺‘((𝑦𝐹)‘𝑑)))
193189, 192eqeq12d 2636 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑) ↔ (𝐺‘((𝑥𝐹)‘𝑑)) = (𝐺‘((𝑦𝐹)‘𝑑))))
19430ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝐺:dom 𝐺1-1-onto𝐵)
195 f1of1 6093 . . . . . . . . . . . . . . . . . . . . 21 (𝐺:𝐵1-1-onto→dom 𝐺𝐺:𝐵1-1→dom 𝐺)
196194, 19, 1953syl 18 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝐺:𝐵1-1→dom 𝐺)
197186, 187ffvelrnd 6316 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑥𝐹)‘𝑑) ∈ 𝐵)
198190, 187ffvelrnd 6316 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑦𝐹)‘𝑑) ∈ 𝐵)
199 f1fveq 6473 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:𝐵1-1→dom 𝐺 ∧ (((𝑥𝐹)‘𝑑) ∈ 𝐵 ∧ ((𝑦𝐹)‘𝑑) ∈ 𝐵)) → ((𝐺‘((𝑥𝐹)‘𝑑)) = (𝐺‘((𝑦𝐹)‘𝑑)) ↔ ((𝑥𝐹)‘𝑑) = ((𝑦𝐹)‘𝑑)))
200196, 197, 198, 199syl12anc 1321 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝐺‘((𝑥𝐹)‘𝑑)) = (𝐺‘((𝑦𝐹)‘𝑑)) ↔ ((𝑥𝐹)‘𝑑) = ((𝑦𝐹)‘𝑑)))
201 fvco3 6232 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:dom 𝐹𝐴𝑑 ∈ dom 𝐹) → ((𝑥𝐹)‘𝑑) = (𝑥‘(𝐹𝑑)))
202136, 187, 201sylancr 694 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑥𝐹)‘𝑑) = (𝑥‘(𝐹𝑑)))
203 fvco3 6232 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:dom 𝐹𝐴𝑑 ∈ dom 𝐹) → ((𝑦𝐹)‘𝑑) = (𝑦‘(𝐹𝑑)))
204136, 187, 203sylancr 694 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑦𝐹)‘𝑑) = (𝑦‘(𝐹𝑑)))
205202, 204eqeq12d 2636 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (((𝑥𝐹)‘𝑑) = ((𝑦𝐹)‘𝑑) ↔ (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑))))
206193, 200, 2053bitrd 294 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑) ↔ (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑))))
207185, 206imbi12d 334 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)) ↔ ((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
208207anassrs 679 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) ∧ 𝑑 ∈ dom 𝐹) → ((𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)) ↔ ((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
209208ralbidva 2979 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
210180, 209bitr4d 271 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑))))
211170, 210anbi12d 746 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ (((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
212211rexbidva 3042 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∃𝑐 ∈ dom 𝐹((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ∃𝑐 ∈ dom 𝐹(((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))))
213115, 127, 2123bitr4rd 301 . . . . . . . . . . 11 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∃𝑐 ∈ dom 𝐹((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)))
21484, 88, 2133bitr3d 298 . . . . . . . . . 10 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)))
215214ex 450 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ((𝑥𝑈𝑦𝑈) → (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦))))
216215pm5.32rd 671 . . . . . . . 8 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ((∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ∧ (𝑥𝑈𝑦𝑈)) ↔ (((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) ∧ (𝑥𝑈𝑦𝑈))))
217216opabbidv 4678 . . . . . . 7 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → {⟨𝑥, 𝑦⟩ ∣ (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ∧ (𝑥𝑈𝑦𝑈))} = {⟨𝑥, 𝑦⟩ ∣ (((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) ∧ (𝑥𝑈𝑦𝑈))})
218 wemapwe.t . . . . . . . . 9 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
219 df-xp 5080 . . . . . . . . 9 (𝑈 × 𝑈) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)}
220218, 219ineq12i 3790 . . . . . . . 8 (𝑇 ∩ (𝑈 × 𝑈)) = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)})
221 inopab 5212 . . . . . . . 8 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)}) = {⟨𝑥, 𝑦⟩ ∣ (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ∧ (𝑥𝑈𝑦𝑈))}
222220, 221eqtri 2643 . . . . . . 7 (𝑇 ∩ (𝑈 × 𝑈)) = {⟨𝑥, 𝑦⟩ ∣ (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ∧ (𝑥𝑈𝑦𝑈))}
223219ineq2i 3789 . . . . . . . 8 ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)})
224 inopab 5212 . . . . . . . 8 ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)}) = {⟨𝑥, 𝑦⟩ ∣ (((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) ∧ (𝑥𝑈𝑦𝑈))}
225223, 224eqtri 2643 . . . . . . 7 ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) = {⟨𝑥, 𝑦⟩ ∣ (((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) ∧ (𝑥𝑈𝑦𝑈))}
226217, 222, 2253eqtr4g 2680 . . . . . 6 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑇 ∩ (𝑈 × 𝑈)) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)))
227 weeq1 5062 . . . . . 6 ((𝑇 ∩ (𝑈 × 𝑈)) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) → ((𝑇 ∩ (𝑈 × 𝑈)) We 𝑈 ↔ ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) We 𝑈))
228226, 227syl 17 . . . . 5 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ((𝑇 ∩ (𝑈 × 𝑈)) We 𝑈 ↔ ({⟨𝑥, 𝑦⟩ ∣ ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥){⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ dom 𝐹((𝑎𝑐) ∈ (𝑏𝑐) ∧ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)))} ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦)} ∩ (𝑈 × 𝑈)) We 𝑈))
22973, 228mpbird 247 . . . 4 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑇 ∩ (𝑈 × 𝑈)) We 𝑈)
230 weinxp 5147 . . . 4 (𝑇 We 𝑈 ↔ (𝑇 ∩ (𝑈 × 𝑈)) We 𝑈)
231229, 230sylibr 224 . . 3 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑇 We 𝑈)
232231ex 450 . 2 (𝜑 → ((𝐵 ∈ V ∧ 𝐴 ∈ V) → 𝑇 We 𝑈))
233 we0 5069 . . 3 𝑇 We ∅
234 elmapex 7822 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑚 𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
235234con3i 150 . . . . . . . 8 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ¬ 𝑥 ∈ (𝐵𝑚 𝐴))
236235pm2.21d 118 . . . . . . 7 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐵𝑚 𝐴) → ¬ 𝑥 finSupp 𝑍))
237236ralrimiv 2959 . . . . . 6 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ∀𝑥 ∈ (𝐵𝑚 𝐴) ¬ 𝑥 finSupp 𝑍)
238 rabeq0 3931 . . . . . 6 ({𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅ ↔ ∀𝑥 ∈ (𝐵𝑚 𝐴) ¬ 𝑥 finSupp 𝑍)
239237, 238sylibr 224 . . . . 5 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → {𝑥 ∈ (𝐵𝑚 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅)
2401, 239syl5eq 2667 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → 𝑈 = ∅)
241 weeq2 5063 . . . 4 (𝑈 = ∅ → (𝑇 We 𝑈𝑇 We ∅))
242240, 241syl 17 . . 3 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝑇 We 𝑈𝑇 We ∅))
243233, 242mpbiri 248 . 2 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → 𝑇 We 𝑈)
244232, 243pm2.61d1 171 1 (𝜑𝑇 We 𝑈)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cin 3554  c0 3891   class class class wbr 4613  {copab 4672  cmpt 4673   E cep 4983   We wwe 5032   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  ccom 5078  Ord word 5681  Oncon0 5682   Fn wfn 5842  wf 5843  1-1wf1 5844  ontowfo 5845  1-1-ontowf1o 5846  cfv 5847   Isom wiso 5848  (class class class)co 6604  𝑜 coe 7504  𝑚 cmap 7802   finSupp cfsupp 8219  OrdIsocoi 8358   CNF ccnf 8502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-1o 7505  df-2o 7506  df-oadd 7509  df-omul 7510  df-oexp 7511  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-oi 8359  df-cnf 8503
This theorem is referenced by:  ltbwe  19391
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