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Theorem wemapwe 9148
 Description: Construct lexicographic order on a function space based on a reverse well-ordering of the indices 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 2801 . . . . . . . . 9 {𝑥 ∈ (dom 𝐺m dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)} = {𝑥 ∈ (dom 𝐺m dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)}
3 eqid 2801 . . . . . . . . 9 (𝐺𝑍) = (𝐺𝑍)
4 simprr 772 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐴 ∈ V)
5 wemapwe.2 . . . . . . . . . . . 12 (𝜑𝑅 We 𝐴)
65adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑅 We 𝐴)
7 wemapwe.5 . . . . . . . . . . . 12 𝐹 = OrdIso(𝑅, 𝐴)
87oiiso 8989 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
94, 6, 8syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
10 isof1o 7059 . . . . . . . . . 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 770 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐵 ∈ V)
13 wemapwe.3 . . . . . . . . . . . 12 (𝜑𝑆 We 𝐵)
1413adantr 484 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑆 We 𝐵)
15 wemapwe.6 . . . . . . . . . . . 12 𝐺 = OrdIso(𝑆, 𝐵)
1615oiiso 8989 . . . . . . . . . . 11 ((𝐵 ∈ V ∧ 𝑆 We 𝐵) → 𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵))
1712, 14, 16syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵))
18 isof1o 7059 . . . . . . . . . 10 (𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵) → 𝐺:dom 𝐺1-1-onto𝐵)
19 f1ocnv 6606 . . . . . . . . . 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 8987 . . . . . . . . . . 11 (𝐴 ∈ V → 𝐹 ∈ V)
2221ad2antll 728 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐹 ∈ V)
2322dmexd 7600 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐹 ∈ V)
2415oiexg 8987 . . . . . . . . . . 11 (𝐵 ∈ V → 𝐺 ∈ V)
2524ad2antrl 727 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐺 ∈ V)
2625dmexd 7600 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐺 ∈ V)
27 wemapwe.7 . . . . . . . . . 10 𝑍 = (𝐺‘∅)
2817, 18syl 17 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐺:dom 𝐺1-1-onto𝐵)
29 f1ofo 6601 . . . . . . . . . . . . . . 15 (𝐺:dom 𝐺1-1-onto𝐵𝐺:dom 𝐺onto𝐵)
30 forn 6572 . . . . . . . . . . . . . . 15 (𝐺:dom 𝐺onto𝐵 → ran 𝐺 = 𝐵)
3128, 29, 303syl 18 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ran 𝐺 = 𝐵)
32 wemapwe.4 . . . . . . . . . . . . . . 15 (𝜑𝐵 ≠ ∅)
3332adantr 484 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝐵 ≠ ∅)
3431, 33eqnetrd 3057 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ran 𝐺 ≠ ∅)
35 dm0rn0 5763 . . . . . . . . . . . . . 14 (dom 𝐺 = ∅ ↔ ran 𝐺 = ∅)
3635necon3bii 3042 . . . . . . . . . . . . 13 (dom 𝐺 ≠ ∅ ↔ ran 𝐺 ≠ ∅)
3734, 36sylibr 237 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐺 ≠ ∅)
3815oicl 8981 . . . . . . . . . . . . 13 Ord dom 𝐺
39 ord0eln0 6217 . . . . . . . . . . . . 13 (Ord dom 𝐺 → (∅ ∈ dom 𝐺 ↔ dom 𝐺 ≠ ∅))
4038, 39ax-mp 5 . . . . . . . . . . . 12 (∅ ∈ dom 𝐺 ↔ dom 𝐺 ≠ ∅)
4137, 40sylibr 237 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → ∅ ∈ dom 𝐺)
4215oif 8982 . . . . . . . . . . . 12 𝐺:dom 𝐺𝐵
4342ffvelrni 6831 . . . . . . . . . . 11 (∅ ∈ dom 𝐺 → (𝐺‘∅) ∈ 𝐵)
4441, 43syl 17 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝐺‘∅) ∈ 𝐵)
4527, 44eqeltrid 2897 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑍𝐵)
461, 2, 3, 11, 20, 4, 12, 23, 26, 45mapfien 8859 . . . . . . . 8 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→{𝑥 ∈ (dom 𝐺m dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)})
47 eqid 2801 . . . . . . . . . . 11 {𝑥 ∈ (dom 𝐺m dom 𝐹) ∣ 𝑥 finSupp ∅} = {𝑥 ∈ (dom 𝐺m dom 𝐹) ∣ 𝑥 finSupp ∅}
4815oion 8988 . . . . . . . . . . . 12 (𝐵 ∈ V → dom 𝐺 ∈ On)
4948ad2antrl 727 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐺 ∈ On)
507oion 8988 . . . . . . . . . . . 12 (𝐴 ∈ V → dom 𝐹 ∈ On)
5150ad2antll 728 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom 𝐹 ∈ On)
5247, 49, 51cantnfdm 9115 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom (dom 𝐺 CNF dom 𝐹) = {𝑥 ∈ (dom 𝐺m dom 𝐹) ∣ 𝑥 finSupp ∅})
5327fveq2i 6652 . . . . . . . . . . . . 13 (𝐺𝑍) = (𝐺‘(𝐺‘∅))
54 f1ocnvfv1 7015 . . . . . . . . . . . . . 14 ((𝐺:dom 𝐺1-1-onto𝐵 ∧ ∅ ∈ dom 𝐺) → (𝐺‘(𝐺‘∅)) = ∅)
5528, 41, 54syl2anc 587 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝐺‘(𝐺‘∅)) = ∅)
5653, 55syl5eq 2848 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝐺𝑍) = ∅)
5756breq2d 5045 . . . . . . . . . . 11 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑥 finSupp (𝐺𝑍) ↔ 𝑥 finSupp ∅))
5857rabbidv 3430 . . . . . . . . . 10 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → {𝑥 ∈ (dom 𝐺m dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)} = {𝑥 ∈ (dom 𝐺m dom 𝐹) ∣ 𝑥 finSupp ∅})
5952, 58eqtr4d 2839 . . . . . . . . 9 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → dom (dom 𝐺 CNF dom 𝐹) = {𝑥 ∈ (dom 𝐺m dom 𝐹) ∣ 𝑥 finSupp (𝐺𝑍)})
6059f1oeq3d 6591 . . . . . . . 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)}))
6146, 60mpbird 260 . . . . . . 7 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))):𝑈1-1-onto→dom (dom 𝐺 CNF dom 𝐹))
62 eqid 2801 . . . . . . . . 9 dom (dom 𝐺 CNF dom 𝐹) = dom (dom 𝐺 CNF dom 𝐹)
63 eqid 2801 . . . . . . . . 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)))}
6462, 49, 51, 63oemapwe 9145 . . . . . . . 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)))
6564simpld 498 . . . . . . 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))
66 eqid 2801 . . . . . . . 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)}
6766f1owe 7089 . . . . . . 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))
6861, 65, 67sylc 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)
69 weinxp 5604 . . . . . 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)
7068, 69sylib 221 . . . . 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)
7111adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐹:dom 𝐹1-1-onto𝐴)
72 f1ofn 6595 . . . . . . . . . . . 12 (𝐹:dom 𝐹1-1-onto𝐴𝐹 Fn dom 𝐹)
73 fveq2 6649 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑐) → (𝑥𝑧) = (𝑥‘(𝐹𝑐)))
74 fveq2 6649 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑐) → (𝑦𝑧) = (𝑦‘(𝐹𝑐)))
7573, 74breq12d 5046 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑐) → ((𝑥𝑧)𝑆(𝑦𝑧) ↔ (𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐))))
76 breq1 5036 . . . . . . . . . . . . . . . 16 (𝑧 = (𝐹𝑐) → (𝑧𝑅𝑤 ↔ (𝐹𝑐)𝑅𝑤))
7776imbi1d 345 . . . . . . . . . . . . . . 15 (𝑧 = (𝐹𝑐) → ((𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))))
7877ralbidv 3165 . . . . . . . . . . . . . 14 (𝑧 = (𝐹𝑐) → (∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))))
7975, 78anbi12d 633 . . . . . . . . . . . . 13 (𝑧 = (𝐹𝑐) → (((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ↔ ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ∧ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))))
8079rexrn 6834 . . . . . . . . . . . 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)))))
8171, 72, 803syl 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)))))
82 f1ofo 6601 . . . . . . . . . . . . 13 (𝐹:dom 𝐹1-1-onto𝐴𝐹:dom 𝐹onto𝐴)
83 forn 6572 . . . . . . . . . . . . 13 (𝐹:dom 𝐹onto𝐴 → ran 𝐹 = 𝐴)
8471, 82, 833syl 18 . . . . . . . . . . . 12 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → ran 𝐹 = 𝐴)
8584rexeqdv 3368 . . . . . . . . . . 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)))))
8625adantr 484 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐺 ∈ V)
87 cnvexg 7615 . . . . . . . . . . . . . . 15 (𝐺 ∈ V → 𝐺 ∈ V)
8886, 87syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐺 ∈ V)
89 vex 3447 . . . . . . . . . . . . . . 15 𝑥 ∈ V
9022adantr 484 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝐹 ∈ V)
91 coexg 7620 . . . . . . . . . . . . . . 15 ((𝑥 ∈ V ∧ 𝐹 ∈ V) → (𝑥𝐹) ∈ V)
9289, 90, 91sylancr 590 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝑥𝐹) ∈ V)
93 coexg 7620 . . . . . . . . . . . . . 14 ((𝐺 ∈ V ∧ (𝑥𝐹) ∈ V) → (𝐺 ∘ (𝑥𝐹)) ∈ V)
9488, 92, 93syl2anc 587 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝐺 ∘ (𝑥𝐹)) ∈ V)
95 vex 3447 . . . . . . . . . . . . . . 15 𝑦 ∈ V
96 coexg 7620 . . . . . . . . . . . . . . 15 ((𝑦 ∈ V ∧ 𝐹 ∈ V) → (𝑦𝐹) ∈ V)
9795, 90, 96sylancr 590 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝑦𝐹) ∈ V)
98 coexg 7620 . . . . . . . . . . . . . 14 ((𝐺 ∈ V ∧ (𝑦𝐹) ∈ V) → (𝐺 ∘ (𝑦𝐹)) ∈ V)
9988, 97, 98syl2anc 587 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (𝐺 ∘ (𝑦𝐹)) ∈ V)
100 fveq1 6648 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝐺 ∘ (𝑥𝐹)) → (𝑎𝑐) = ((𝐺 ∘ (𝑥𝐹))‘𝑐))
101 fveq1 6648 . . . . . . . . . . . . . . . . 17 (𝑏 = (𝐺 ∘ (𝑦𝐹)) → (𝑏𝑐) = ((𝐺 ∘ (𝑦𝐹))‘𝑐))
102 eleq12 2882 . . . . . . . . . . . . . . . . 17 (((𝑎𝑐) = ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∧ (𝑏𝑐) = ((𝐺 ∘ (𝑦𝐹))‘𝑐)) → ((𝑎𝑐) ∈ (𝑏𝑐) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐)))
103100, 101, 102syl2an 598 . . . . . . . . . . . . . . . 16 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → ((𝑎𝑐) ∈ (𝑏𝑐) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐)))
104 fveq1 6648 . . . . . . . . . . . . . . . . . . 19 (𝑎 = (𝐺 ∘ (𝑥𝐹)) → (𝑎𝑑) = ((𝐺 ∘ (𝑥𝐹))‘𝑑))
105 fveq1 6648 . . . . . . . . . . . . . . . . . . 19 (𝑏 = (𝐺 ∘ (𝑦𝐹)) → (𝑏𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑))
106104, 105eqeqan12d 2818 . . . . . . . . . . . . . . . . . 18 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → ((𝑎𝑑) = (𝑏𝑑) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑)))
107106imbi2d 344 . . . . . . . . . . . . . . . . 17 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → ((𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)) ↔ (𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑))))
108107ralbidv 3165 . . . . . . . . . . . . . . . 16 ((𝑎 = (𝐺 ∘ (𝑥𝐹)) ∧ 𝑏 = (𝐺 ∘ (𝑦𝐹))) → (∀𝑑 ∈ dom 𝐹(𝑐𝑑 → (𝑎𝑑) = (𝑏𝑑)) ↔ ∀𝑑 ∈ dom 𝐹(𝑐𝑑 → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑))))
109103, 108anbi12d 633 . . . . . . . . . . . . . . 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)))))
110109rexbidv 3259 . . . . . . . . . . . . . 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)))))
111110, 63brabga 5389 . . . . . . . . . . . . 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)))))
11294, 99, 111syl2anc 587 . . . . . . . . . . . 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)))))
113 eqid 2801 . . . . . . . . . . . . . 14 (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹))) = (𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))
114 coeq1 5696 . . . . . . . . . . . . . . 15 (𝑓 = 𝑥 → (𝑓𝐹) = (𝑥𝐹))
115114coeq2d 5701 . . . . . . . . . . . . . 14 (𝑓 = 𝑥 → (𝐺 ∘ (𝑓𝐹)) = (𝐺 ∘ (𝑥𝐹)))
116 simprl 770 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑥𝑈)
117113, 115, 116, 94fvmptd3 6772 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑥) = (𝐺 ∘ (𝑥𝐹)))
118 coeq1 5696 . . . . . . . . . . . . . . 15 (𝑓 = 𝑦 → (𝑓𝐹) = (𝑦𝐹))
119118coeq2d 5701 . . . . . . . . . . . . . 14 (𝑓 = 𝑦 → (𝐺 ∘ (𝑓𝐹)) = (𝐺 ∘ (𝑦𝐹)))
120 simprr 772 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑦𝑈)
121113, 119, 120, 99fvmptd3 6772 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → ((𝑓𝑈 ↦ (𝐺 ∘ (𝑓𝐹)))‘𝑦) = (𝐺 ∘ (𝑦𝐹)))
122117, 121breq12d 5046 . . . . . . . . . . . 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))))
12317ad2antrr 725 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵))
124 isocnv 7066 . . . . . . . . . . . . . . . . . 18 (𝐺 Isom E , 𝑆 (dom 𝐺, 𝐵) → 𝐺 Isom 𝑆, E (𝐵, dom 𝐺))
125123, 124syl 17 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝐺 Isom 𝑆, E (𝐵, dom 𝐺))
1261ssrab3 4011 . . . . . . . . . . . . . . . . . . . 20 𝑈 ⊆ (𝐵m 𝐴)
127126, 116sseldi 3916 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑥 ∈ (𝐵m 𝐴))
128 elmapi 8415 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ (𝐵m 𝐴) → 𝑥:𝐴𝐵)
129127, 128syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑥:𝐴𝐵)
1307oif 8982 . . . . . . . . . . . . . . . . . . 19 𝐹:dom 𝐹𝐴
131130ffvelrni 6831 . . . . . . . . . . . . . . . . . 18 (𝑐 ∈ dom 𝐹 → (𝐹𝑐) ∈ 𝐴)
132 ffvelrn 6830 . . . . . . . . . . . . . . . . . 18 ((𝑥:𝐴𝐵 ∧ (𝐹𝑐) ∈ 𝐴) → (𝑥‘(𝐹𝑐)) ∈ 𝐵)
133129, 131, 132syl2an 598 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑥‘(𝐹𝑐)) ∈ 𝐵)
134126, 120sseldi 3916 . . . . . . . . . . . . . . . . . . 19 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑦 ∈ (𝐵m 𝐴))
135 elmapi 8415 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ (𝐵m 𝐴) → 𝑦:𝐴𝐵)
136134, 135syl 17 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → 𝑦:𝐴𝐵)
137 ffvelrn 6830 . . . . . . . . . . . . . . . . . 18 ((𝑦:𝐴𝐵 ∧ (𝐹𝑐) ∈ 𝐴) → (𝑦‘(𝐹𝑐)) ∈ 𝐵)
138136, 131, 137syl2an 598 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑦‘(𝐹𝑐)) ∈ 𝐵)
139 isorel 7062 . . . . . . . . . . . . . . . . 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)))))
140125, 133, 138, 139syl12anc 835 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) E (𝐺‘(𝑦‘(𝐹𝑐)))))
141 fvex 6662 . . . . . . . . . . . . . . . . 17 (𝐺‘(𝑦‘(𝐹𝑐))) ∈ V
142141epeli 5436 . . . . . . . . . . . . . . . 16 ((𝐺‘(𝑥‘(𝐹𝑐))) E (𝐺‘(𝑦‘(𝐹𝑐))) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) ∈ (𝐺‘(𝑦‘(𝐹𝑐))))
143140, 142syl6bb 290 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) ∈ (𝐺‘(𝑦‘(𝐹𝑐)))))
144129adantr 484 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝑥:𝐴𝐵)
145 fco 6509 . . . . . . . . . . . . . . . . . . 19 ((𝑥:𝐴𝐵𝐹:dom 𝐹𝐴) → (𝑥𝐹):dom 𝐹𝐵)
146144, 130, 145sylancl 589 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑥𝐹):dom 𝐹𝐵)
147 fvco3 6741 . . . . . . . . . . . . . . . . . 18 (((𝑥𝐹):dom 𝐹𝐵𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑐) = (𝐺‘((𝑥𝐹)‘𝑐)))
148146, 147sylancom 591 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑐) = (𝐺‘((𝑥𝐹)‘𝑐)))
149 simpr 488 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝑐 ∈ dom 𝐹)
150 fvco3 6741 . . . . . . . . . . . . . . . . . . 19 ((𝐹:dom 𝐹𝐴𝑐 ∈ dom 𝐹) → ((𝑥𝐹)‘𝑐) = (𝑥‘(𝐹𝑐)))
151130, 149, 150sylancr 590 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥𝐹)‘𝑐) = (𝑥‘(𝐹𝑐)))
152151fveq2d 6653 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝐺‘((𝑥𝐹)‘𝑐)) = (𝐺‘(𝑥‘(𝐹𝑐))))
153148, 152eqtrd 2836 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑐) = (𝐺‘(𝑥‘(𝐹𝑐))))
154136adantr 484 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → 𝑦:𝐴𝐵)
155 fco 6509 . . . . . . . . . . . . . . . . . . 19 ((𝑦:𝐴𝐵𝐹:dom 𝐹𝐴) → (𝑦𝐹):dom 𝐹𝐵)
156154, 130, 155sylancl 589 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝑦𝐹):dom 𝐹𝐵)
157 fvco3 6741 . . . . . . . . . . . . . . . . . 18 (((𝑦𝐹):dom 𝐹𝐵𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑐) = (𝐺‘((𝑦𝐹)‘𝑐)))
158156, 157sylancom 591 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑐) = (𝐺‘((𝑦𝐹)‘𝑐)))
159 fvco3 6741 . . . . . . . . . . . . . . . . . . 19 ((𝐹:dom 𝐹𝐴𝑐 ∈ dom 𝐹) → ((𝑦𝐹)‘𝑐) = (𝑦‘(𝐹𝑐)))
160130, 149, 159sylancr 590 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑦𝐹)‘𝑐) = (𝑦‘(𝐹𝑐)))
161160fveq2d 6653 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (𝐺‘((𝑦𝐹)‘𝑐)) = (𝐺‘(𝑦‘(𝐹𝑐))))
162158, 161eqtrd 2836 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑐) = (𝐺‘(𝑦‘(𝐹𝑐))))
163153, 162eleq12d 2887 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → (((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐) ↔ (𝐺‘(𝑥‘(𝐹𝑐))) ∈ (𝐺‘(𝑦‘(𝐹𝑐)))))
164143, 163bitr4d 285 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ 𝑐 ∈ dom 𝐹) → ((𝑥‘(𝐹𝑐))𝑆(𝑦‘(𝐹𝑐)) ↔ ((𝐺 ∘ (𝑥𝐹))‘𝑐) ∈ ((𝐺 ∘ (𝑦𝐹))‘𝑐)))
16584raleqdv 3367 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∀𝑤 ∈ ran 𝐹((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))))
166 breq2 5037 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐹𝑑) → ((𝐹𝑐)𝑅𝑤 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
167 fveq2 6649 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐹𝑑) → (𝑥𝑤) = (𝑥‘(𝐹𝑑)))
168 fveq2 6649 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = (𝐹𝑑) → (𝑦𝑤) = (𝑦‘(𝐹𝑑)))
169167, 168eqeq12d 2817 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = (𝐹𝑑) → ((𝑥𝑤) = (𝑦𝑤) ↔ (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑))))
170166, 169imbi12d 348 . . . . . . . . . . . . . . . . . . 19 (𝑤 = (𝐹𝑑) → (((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
171170ralrn 6835 . . . . . . . . . . . . . . . . . 18 (𝐹 Fn dom 𝐹 → (∀𝑤 ∈ ran 𝐹((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
17271, 72, 1713syl 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)))))
173165, 172bitr3d 284 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) → (∀𝑤𝐴 ((𝐹𝑐)𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)) ↔ ∀𝑑 ∈ dom 𝐹((𝐹𝑐)𝑅(𝐹𝑑) → (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑)))))
174173adantr 484 . . . . . . . . . . . . . . 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)))))
175 epel 5437 . . . . . . . . . . . . . . . . . . 19 (𝑐 E 𝑑𝑐𝑑)
1769ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴))
177 isorel 7062 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑐 E 𝑑 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
178176, 177sylancom 591 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑐 E 𝑑 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
179175, 178bitr3id 288 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑐𝑑 ↔ (𝐹𝑐)𝑅(𝐹𝑑)))
180146adantrr 716 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑥𝐹):dom 𝐹𝐵)
181 simprr 772 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝑑 ∈ dom 𝐹)
182 fvco3 6741 . . . . . . . . . . . . . . . . . . . . 21 (((𝑥𝐹):dom 𝐹𝐵𝑑 ∈ dom 𝐹) → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = (𝐺‘((𝑥𝐹)‘𝑑)))
183180, 181, 182syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝐺 ∘ (𝑥𝐹))‘𝑑) = (𝐺‘((𝑥𝐹)‘𝑑)))
184156adantrr 716 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (𝑦𝐹):dom 𝐹𝐵)
185 fvco3 6741 . . . . . . . . . . . . . . . . . . . . 21 (((𝑦𝐹):dom 𝐹𝐵𝑑 ∈ dom 𝐹) → ((𝐺 ∘ (𝑦𝐹))‘𝑑) = (𝐺‘((𝑦𝐹)‘𝑑)))
186184, 181, 185syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝐺 ∘ (𝑦𝐹))‘𝑑) = (𝐺‘((𝑦𝐹)‘𝑑)))
187183, 186eqeq12d 2817 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑) ↔ (𝐺‘((𝑥𝐹)‘𝑑)) = (𝐺‘((𝑦𝐹)‘𝑑))))
18828ad2antrr 725 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝐺:dom 𝐺1-1-onto𝐵)
189 f1of1 6593 . . . . . . . . . . . . . . . . . . . . 21 (𝐺:𝐵1-1-onto→dom 𝐺𝐺:𝐵1-1→dom 𝐺)
190188, 19, 1893syl 18 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → 𝐺:𝐵1-1→dom 𝐺)
191180, 181ffvelrnd 6833 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑥𝐹)‘𝑑) ∈ 𝐵)
192184, 181ffvelrnd 6833 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑦𝐹)‘𝑑) ∈ 𝐵)
193 f1fveq 7002 . . . . . . . . . . . . . . . . . . . 20 ((𝐺:𝐵1-1→dom 𝐺 ∧ (((𝑥𝐹)‘𝑑) ∈ 𝐵 ∧ ((𝑦𝐹)‘𝑑) ∈ 𝐵)) → ((𝐺‘((𝑥𝐹)‘𝑑)) = (𝐺‘((𝑦𝐹)‘𝑑)) ↔ ((𝑥𝐹)‘𝑑) = ((𝑦𝐹)‘𝑑)))
194190, 191, 192, 193syl12anc 835 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝐺‘((𝑥𝐹)‘𝑑)) = (𝐺‘((𝑦𝐹)‘𝑑)) ↔ ((𝑥𝐹)‘𝑑) = ((𝑦𝐹)‘𝑑)))
195 fvco3 6741 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:dom 𝐹𝐴𝑑 ∈ dom 𝐹) → ((𝑥𝐹)‘𝑑) = (𝑥‘(𝐹𝑑)))
196130, 181, 195sylancr 590 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑥𝐹)‘𝑑) = (𝑥‘(𝐹𝑑)))
197 fvco3 6741 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹:dom 𝐹𝐴𝑑 ∈ dom 𝐹) → ((𝑦𝐹)‘𝑑) = (𝑦‘(𝐹𝑑)))
198130, 181, 197sylancr 590 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → ((𝑦𝐹)‘𝑑) = (𝑦‘(𝐹𝑑)))
199196, 198eqeq12d 2817 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (((𝑥𝐹)‘𝑑) = ((𝑦𝐹)‘𝑑) ↔ (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑))))
200187, 194, 1993bitrd 308 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) ∧ (𝑥𝑈𝑦𝑈)) ∧ (𝑐 ∈ dom 𝐹𝑑 ∈ dom 𝐹)) → (((𝐺 ∘ (𝑥𝐹))‘𝑑) = ((𝐺 ∘ (𝑦𝐹))‘𝑑) ↔ (𝑥‘(𝐹𝑑)) = (𝑦‘(𝐹𝑑))))
201179, 200imbi12d 348 . . . . . . . . . . . . . . . . 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)))))
202201anassrs 471 . . . . . . . . . . . . . . . 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)))))
203202ralbidva 3164 . . . . . . . . . . . . . . 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)))))
204174, 203bitr4d 285 . . . . . . . . . . . . . 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))))
205164, 204anbi12d 633 . . . . . . . . . . . . 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)))))
206205rexbidva 3258 . . . . . . . . . . . 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)))))
207112, 122, 2063bitr4rd 315 . . . . . . . . . . 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)))
20881, 85, 2073bitr3d 312 . . . . . . . . . 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)))
209208ex 416 . . . . . . . . 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))))
210209pm5.32rd 581 . . . . . . . 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))))
211210opabbidv 5099 . . . . . . 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))})
212 wemapwe.t . . . . . . . . 9 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
213 df-xp 5529 . . . . . . . . 9 (𝑈 × 𝑈) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)}
214212, 213ineq12i 4140 . . . . . . . 8 (𝑇 ∩ (𝑈 × 𝑈)) = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤)))} ∩ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝑈𝑦𝑈)})
215 inopab 5669 . . . . . . . 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))}
216214, 215eqtri 2824 . . . . . . 7 (𝑇 ∩ (𝑈 × 𝑈)) = {⟨𝑥, 𝑦⟩ ∣ (∃𝑧𝐴 ((𝑥𝑧)𝑆(𝑦𝑧) ∧ ∀𝑤𝐴 (𝑧𝑅𝑤 → (𝑥𝑤) = (𝑦𝑤))) ∧ (𝑥𝑈𝑦𝑈))}
217213ineq2i 4139 . . . . . . . 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)})
218 inopab 5669 . . . . . . . 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))}
219217, 218eqtri 2824 . . . . . . 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))}
220211, 216, 2193eqtr4g 2861 . . . . . 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)))
221 weeq1 5511 . . . . . 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))
222220, 221syl 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))
22370, 222mpbird 260 . . . 4 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → (𝑇 ∩ (𝑈 × 𝑈)) We 𝑈)
224 weinxp 5604 . . . 4 (𝑇 We 𝑈 ↔ (𝑇 ∩ (𝑈 × 𝑈)) We 𝑈)
225223, 224sylibr 237 . . 3 ((𝜑 ∧ (𝐵 ∈ V ∧ 𝐴 ∈ V)) → 𝑇 We 𝑈)
226225ex 416 . 2 (𝜑 → ((𝐵 ∈ V ∧ 𝐴 ∈ V) → 𝑇 We 𝑈))
227 we0 5518 . . 3 𝑇 We ∅
228 elmapex 8414 . . . . . . . . 9 (𝑥 ∈ (𝐵m 𝐴) → (𝐵 ∈ V ∧ 𝐴 ∈ V))
229228con3i 157 . . . . . . . 8 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ¬ 𝑥 ∈ (𝐵m 𝐴))
230229pm2.21d 121 . . . . . . 7 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ (𝐵m 𝐴) → ¬ 𝑥 finSupp 𝑍))
231230ralrimiv 3151 . . . . . 6 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ∀𝑥 ∈ (𝐵m 𝐴) ¬ 𝑥 finSupp 𝑍)
232 rabeq0 4295 . . . . . 6 ({𝑥 ∈ (𝐵m 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅ ↔ ∀𝑥 ∈ (𝐵m 𝐴) ¬ 𝑥 finSupp 𝑍)
233231, 232sylibr 237 . . . . 5 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → {𝑥 ∈ (𝐵m 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅)
2341, 233syl5eq 2848 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → 𝑈 = ∅)
235 weeq2 5512 . . . 4 (𝑈 = ∅ → (𝑇 We 𝑈𝑇 We ∅))
236234, 235syl 17 . . 3 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝑇 We 𝑈𝑇 We ∅))
237227, 236mpbiri 261 . 2 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → 𝑇 We 𝑈)
238226, 237pm2.61d1 183 1 (𝜑𝑇 We 𝑈)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109  ∃wrex 3110  {crab 3113  Vcvv 3444   ∩ cin 3883  ∅c0 4246   class class class wbr 5033  {copab 5095   ↦ cmpt 5113   E cep 5432   We wwe 5481   × cxp 5521  ◡ccnv 5522  dom cdm 5523  ran crn 5524   ∘ ccom 5527  Ord word 6162  Oncon0 6163   Fn wfn 6323  ⟶wf 6324  –1-1→wf1 6325  –onto→wfo 6326  –1-1-onto→wf1o 6327  ‘cfv 6328   Isom wiso 6329  (class class class)co 7139   ↑o coe 8088   ↑m cmap 8393   finSupp cfsupp 8821  OrdIsocoi 8961   CNF ccnf 9112 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-supp 7818  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-seqom 8071  df-1o 8089  df-2o 8090  df-oadd 8093  df-omul 8094  df-oexp 8095  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-fsupp 8822  df-oi 8962  df-cnf 9113 This theorem is referenced by:  ltbwe  20715
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