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Theorem tfr1onlemsucaccv 6088
Description: Lemma for tfr1on 6097. We can extend an acceptable function by one element to produce an acceptable function. (Contributed by Jim Kingdon, 12-Mar-2022.)
Hypotheses
Ref Expression
tfr1on.f 𝐹 = recs(𝐺)
tfr1on.g (𝜑 → Fun 𝐺)
tfr1on.x (𝜑 → Ord 𝑋)
tfr1on.ex ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
tfr1onlemsucfn.1 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
tfr1onlemsucaccv.yx (𝜑𝑌𝑋)
tfr1onlemsucaccv.zy (𝜑𝑧𝑌)
tfr1onlemsucaccv.u ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
tfr1onlemsucaccv.gfn (𝜑𝑔 Fn 𝑧)
tfr1onlemsucaccv.gacc (𝜑𝑔𝐴)
Assertion
Ref Expression
tfr1onlemsucaccv (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
Distinct variable groups:   𝑓,𝐺,𝑥,𝑦   𝑓,𝑋,𝑥   𝑓,𝑔,𝑥,𝑦   𝜑,𝑓,𝑥   𝑧,𝑓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑔)   𝐴(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐹(𝑥,𝑦,𝑧,𝑓,𝑔)   𝐺(𝑧,𝑔)   𝑋(𝑦,𝑧,𝑔)   𝑌(𝑥,𝑦,𝑧,𝑓,𝑔)

Proof of Theorem tfr1onlemsucaccv
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suceq 4220 . . . . 5 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
21eleq1d 2156 . . . 4 (𝑥 = 𝑧 → (suc 𝑥𝑋 ↔ suc 𝑧𝑋))
3 tfr1onlemsucaccv.u . . . . 5 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
43ralrimiva 2446 . . . 4 (𝜑 → ∀𝑥 𝑋 suc 𝑥𝑋)
5 tfr1onlemsucaccv.zy . . . . 5 (𝜑𝑧𝑌)
6 tfr1onlemsucaccv.yx . . . . 5 (𝜑𝑌𝑋)
7 elunii 3653 . . . . 5 ((𝑧𝑌𝑌𝑋) → 𝑧 𝑋)
85, 6, 7syl2anc 403 . . . 4 (𝜑𝑧 𝑋)
92, 4, 8rspcdva 2727 . . 3 (𝜑 → suc 𝑧𝑋)
10 tfr1on.f . . . 4 𝐹 = recs(𝐺)
11 tfr1on.g . . . 4 (𝜑 → Fun 𝐺)
12 tfr1on.x . . . 4 (𝜑 → Ord 𝑋)
13 tfr1on.ex . . . 4 ((𝜑𝑥𝑋𝑓 Fn 𝑥) → (𝐺𝑓) ∈ V)
14 tfr1onlemsucfn.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥𝑋 (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
155, 6jca 300 . . . . 5 (𝜑 → (𝑧𝑌𝑌𝑋))
16 ordtr1 4206 . . . . 5 (Ord 𝑋 → ((𝑧𝑌𝑌𝑋) → 𝑧𝑋))
1712, 15, 16sylc 61 . . . 4 (𝜑𝑧𝑋)
18 tfr1onlemsucaccv.gfn . . . 4 (𝜑𝑔 Fn 𝑧)
19 tfr1onlemsucaccv.gacc . . . 4 (𝜑𝑔𝐴)
2010, 11, 12, 13, 14, 17, 18, 19tfr1onlemsucfn 6087 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
21 vex 2622 . . . . . 6 𝑢 ∈ V
2221elsuc 4224 . . . . 5 (𝑢 ∈ suc 𝑧 ↔ (𝑢𝑧𝑢 = 𝑧))
23 vex 2622 . . . . . . . . . . 11 𝑔 ∈ V
2414tfr1onlem3ag 6084 . . . . . . . . . . 11 (𝑔 ∈ V → (𝑔𝐴 ↔ ∃𝑣𝑋 (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
2523, 24ax-mp 7 . . . . . . . . . 10 (𝑔𝐴 ↔ ∃𝑣𝑋 (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
2619, 25sylib 120 . . . . . . . . 9 (𝜑 → ∃𝑣𝑋 (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
27 simprrr 507 . . . . . . . . . 10 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))
28 simprrl 506 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → 𝑔 Fn 𝑣)
2918adantr 270 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → 𝑔 Fn 𝑧)
30 fndmu 5101 . . . . . . . . . . . 12 ((𝑔 Fn 𝑣𝑔 Fn 𝑧) → 𝑣 = 𝑧)
3128, 29, 30syl2anc 403 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → 𝑣 = 𝑧)
3231raleqdv 2568 . . . . . . . . . 10 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → (∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
3327, 32mpbid 145 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))
3426, 33rexlimddv 2493 . . . . . . . 8 (𝜑 → ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))
3534r19.21bi 2461 . . . . . . 7 ((𝜑𝑢𝑧) → (𝑔𝑢) = (𝐺‘(𝑔𝑢)))
36 ordelon 4201 . . . . . . . . . . . . 13 ((Ord 𝑋𝑧𝑋) → 𝑧 ∈ On)
3712, 17, 36syl2anc 403 . . . . . . . . . . . 12 (𝜑𝑧 ∈ On)
38 onelon 4202 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑢𝑧) → 𝑢 ∈ On)
3937, 38sylan 277 . . . . . . . . . . 11 ((𝜑𝑢𝑧) → 𝑢 ∈ On)
40 eloni 4193 . . . . . . . . . . 11 (𝑢 ∈ On → Ord 𝑢)
41 ordirr 4348 . . . . . . . . . . 11 (Ord 𝑢 → ¬ 𝑢𝑢)
4239, 40, 413syl 17 . . . . . . . . . 10 ((𝜑𝑢𝑧) → ¬ 𝑢𝑢)
43 elequ2 1648 . . . . . . . . . . . 12 (𝑧 = 𝑢 → (𝑢𝑧𝑢𝑢))
4443biimpcd 157 . . . . . . . . . . 11 (𝑢𝑧 → (𝑧 = 𝑢𝑢𝑢))
4544adantl 271 . . . . . . . . . 10 ((𝜑𝑢𝑧) → (𝑧 = 𝑢𝑢𝑢))
4642, 45mtod 624 . . . . . . . . 9 ((𝜑𝑢𝑧) → ¬ 𝑧 = 𝑢)
4746neqned 2262 . . . . . . . 8 ((𝜑𝑢𝑧) → 𝑧𝑢)
48 fvunsng 5475 . . . . . . . 8 ((𝑢 ∈ V ∧ 𝑧𝑢) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝑔𝑢))
4921, 47, 48sylancr 405 . . . . . . 7 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝑔𝑢))
50 eloni 4193 . . . . . . . . . . . 12 (𝑧 ∈ On → Ord 𝑧)
5137, 50syl 14 . . . . . . . . . . 11 (𝜑 → Ord 𝑧)
52 ordelss 4197 . . . . . . . . . . 11 ((Ord 𝑧𝑢𝑧) → 𝑢𝑧)
5351, 52sylan 277 . . . . . . . . . 10 ((𝜑𝑢𝑧) → 𝑢𝑧)
54 resabs1 4729 . . . . . . . . . 10 (𝑢𝑧 → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢))
5553, 54syl 14 . . . . . . . . 9 ((𝜑𝑢𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢))
56 ordirr 4348 . . . . . . . . . . . . 13 (Ord 𝑧 → ¬ 𝑧𝑧)
5751, 56syl 14 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑧𝑧)
58 fsnunres 5482 . . . . . . . . . . . 12 ((𝑔 Fn 𝑧 ∧ ¬ 𝑧𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) = 𝑔)
5918, 57, 58syl2anc 403 . . . . . . . . . . 11 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) = 𝑔)
6059reseq1d 4700 . . . . . . . . . 10 (𝜑 → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔𝑢))
6160adantr 270 . . . . . . . . 9 ((𝜑𝑢𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔𝑢))
6255, 61eqtr3d 2122 . . . . . . . 8 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢) = (𝑔𝑢))
6362fveq2d 5293 . . . . . . 7 ((𝜑𝑢𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)) = (𝐺‘(𝑔𝑢)))
6435, 49, 633eqtr4d 2130 . . . . . 6 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
65 fneq2 5089 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
6665imbi1d 229 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
6766albidv 1752 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
68133expia 1145 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
6968alrimiv 1802 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
7069ralrimiva 2446 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
7167, 70, 17rspcdva 2727 . . . . . . . . . 10 (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
72 fneq1 5088 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
73 fveq2 5289 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
7473eleq1d 2156 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
7572, 74imbi12d 232 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
7675spv 1788 . . . . . . . . . 10 (∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
7771, 18, 76sylc 61 . . . . . . . . 9 (𝜑 → (𝐺𝑔) ∈ V)
78 fndm 5099 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
7918, 78syl 14 . . . . . . . . . 10 (𝜑 → dom 𝑔 = 𝑧)
8057, 79neleqtrrd 2186 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
81 fsnunfv 5481 . . . . . . . . 9 ((𝑧𝑌 ∧ (𝐺𝑔) ∈ V ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧) = (𝐺𝑔))
825, 77, 80, 81syl3anc 1174 . . . . . . . 8 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧) = (𝐺𝑔))
8382adantr 270 . . . . . . 7 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧) = (𝐺𝑔))
84 simpr 108 . . . . . . . 8 ((𝜑𝑢 = 𝑧) → 𝑢 = 𝑧)
8584fveq2d 5293 . . . . . . 7 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧))
86 reseq2 4696 . . . . . . . . 9 (𝑢 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧))
8786, 59sylan9eqr 2142 . . . . . . . 8 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢) = 𝑔)
8887fveq2d 5293 . . . . . . 7 ((𝜑𝑢 = 𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)) = (𝐺𝑔))
8983, 85, 883eqtr4d 2130 . . . . . 6 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
9064, 89jaodan 746 . . . . 5 ((𝜑 ∧ (𝑢𝑧𝑢 = 𝑧)) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
9122, 90sylan2b 281 . . . 4 ((𝜑𝑢 ∈ suc 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
9291ralrimiva 2446 . . 3 (𝜑 → ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
93 fneq2 5089 . . . . 5 (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn 𝑤 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧))
94 raleq 2562 . . . . 5 (𝑤 = suc 𝑧 → (∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)) ↔ ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢))))
9593, 94anbi12d 457 . . . 4 (𝑤 = suc 𝑧 → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢))) ↔ ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧 ∧ ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))))
9695rspcev 2722 . . 3 ((suc 𝑧𝑋 ∧ ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧 ∧ ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))) → ∃𝑤𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢))))
979, 20, 92, 96syl12anc 1172 . 2 (𝜑 → ∃𝑤𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢))))
98 vex 2622 . . . . . 6 𝑧 ∈ V
99 opexg 4046 . . . . . 6 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
10098, 77, 99sylancr 405 . . . . 5 (𝜑 → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
101 snexg 4010 . . . . 5 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
102100, 101syl 14 . . . 4 (𝜑 → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
103 unexg 4259 . . . 4 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
10423, 102, 103sylancr 405 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
10514tfr1onlem3ag 6084 . . 3 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴 ↔ ∃𝑤𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))))
106104, 105syl 14 . 2 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴 ↔ ∃𝑤𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))))
10797, 106mpbird 165 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664  w3a 924  wal 1287   = wceq 1289  wcel 1438  {cab 2074  wne 2255  wral 2359  wrex 2360  Vcvv 2619  cun 2995  wss 2997  {csn 3441  cop 3444   cuni 3648  Ord word 4180  Oncon0 4181  suc csuc 4183  dom cdm 4428  cres 4430  Fun wfun 4996   Fn wfn 4997  cfv 5002  recscrecs 6051
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-res 4440  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010
This theorem is referenced by:  tfr1onlembacc  6089  tfr1onlemres  6096
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