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Theorem tfr1onlemsucaccv 6285
 Description: Lemma for tfr1on 6294. 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 4362 . . . . 5 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
21eleq1d 2226 . . . 4 (𝑥 = 𝑧 → (suc 𝑥𝑋 ↔ suc 𝑧𝑋))
3 tfr1onlemsucaccv.u . . . . 5 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
43ralrimiva 2530 . . . 4 (𝜑 → ∀𝑥 𝑋 suc 𝑥𝑋)
5 tfr1onlemsucaccv.zy . . . . 5 (𝜑𝑧𝑌)
6 tfr1onlemsucaccv.yx . . . . 5 (𝜑𝑌𝑋)
7 elunii 3777 . . . . 5 ((𝑧𝑌𝑌𝑋) → 𝑧 𝑋)
85, 6, 7syl2anc 409 . . . 4 (𝜑𝑧 𝑋)
92, 4, 8rspcdva 2821 . . 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 304 . . . . 5 (𝜑 → (𝑧𝑌𝑌𝑋))
16 ordtr1 4348 . . . . 5 (Ord 𝑋 → ((𝑧𝑌𝑌𝑋) → 𝑧𝑋))
1712, 15, 16sylc 62 . . . 4 (𝜑𝑧𝑋)
18 tfr1onlemsucaccv.gfn . . . 4 (𝜑𝑔 Fn 𝑧)
19 tfr1onlemsucaccv.gacc . . . 4 (𝜑𝑔𝐴)
2010, 11, 12, 13, 14, 17, 18, 19tfr1onlemsucfn 6284 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
21 vex 2715 . . . . . 6 𝑢 ∈ V
2221elsuc 4366 . . . . 5 (𝑢 ∈ suc 𝑧 ↔ (𝑢𝑧𝑢 = 𝑧))
23 vex 2715 . . . . . . . . . . 11 𝑔 ∈ V
2414tfr1onlem3ag 6281 . . . . . . . . . . 11 (𝑔 ∈ V → (𝑔𝐴 ↔ ∃𝑣𝑋 (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))))
2523, 24ax-mp 5 . . . . . . . . . 10 (𝑔𝐴 ↔ ∃𝑣𝑋 (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
2619, 25sylib 121 . . . . . . . . 9 (𝜑 → ∃𝑣𝑋 (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
27 simprrr 530 . . . . . . . . . 10 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))
28 simprrl 529 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → 𝑔 Fn 𝑣)
2918adantr 274 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → 𝑔 Fn 𝑧)
30 fndmu 5270 . . . . . . . . . . . 12 ((𝑔 Fn 𝑣𝑔 Fn 𝑧) → 𝑣 = 𝑧)
3128, 29, 30syl2anc 409 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → 𝑣 = 𝑧)
3231raleqdv 2658 . . . . . . . . . 10 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → (∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))
3327, 32mpbid 146 . . . . . . . . 9 ((𝜑 ∧ (𝑣𝑋 ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐺‘(𝑔𝑢))))) → ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))
3426, 33rexlimddv 2579 . . . . . . . 8 (𝜑 → ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))
3534r19.21bi 2545 . . . . . . 7 ((𝜑𝑢𝑧) → (𝑔𝑢) = (𝐺‘(𝑔𝑢)))
36 ordelon 4343 . . . . . . . . . . . . 13 ((Ord 𝑋𝑧𝑋) → 𝑧 ∈ On)
3712, 17, 36syl2anc 409 . . . . . . . . . . . 12 (𝜑𝑧 ∈ On)
38 onelon 4344 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑢𝑧) → 𝑢 ∈ On)
3937, 38sylan 281 . . . . . . . . . . 11 ((𝜑𝑢𝑧) → 𝑢 ∈ On)
40 eloni 4335 . . . . . . . . . . 11 (𝑢 ∈ On → Ord 𝑢)
41 ordirr 4500 . . . . . . . . . . 11 (Ord 𝑢 → ¬ 𝑢𝑢)
4239, 40, 413syl 17 . . . . . . . . . 10 ((𝜑𝑢𝑧) → ¬ 𝑢𝑢)
43 elequ2 2133 . . . . . . . . . . . 12 (𝑧 = 𝑢 → (𝑢𝑧𝑢𝑢))
4443biimpcd 158 . . . . . . . . . . 11 (𝑢𝑧 → (𝑧 = 𝑢𝑢𝑢))
4544adantl 275 . . . . . . . . . 10 ((𝜑𝑢𝑧) → (𝑧 = 𝑢𝑢𝑢))
4642, 45mtod 653 . . . . . . . . 9 ((𝜑𝑢𝑧) → ¬ 𝑧 = 𝑢)
4746neqned 2334 . . . . . . . 8 ((𝜑𝑢𝑧) → 𝑧𝑢)
48 fvunsng 5660 . . . . . . . 8 ((𝑢 ∈ V ∧ 𝑧𝑢) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝑔𝑢))
4921, 47, 48sylancr 411 . . . . . . 7 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝑔𝑢))
50 eloni 4335 . . . . . . . . . . . 12 (𝑧 ∈ On → Ord 𝑧)
5137, 50syl 14 . . . . . . . . . . 11 (𝜑 → Ord 𝑧)
52 ordelss 4339 . . . . . . . . . . 11 ((Ord 𝑧𝑢𝑧) → 𝑢𝑧)
5351, 52sylan 281 . . . . . . . . . 10 ((𝜑𝑢𝑧) → 𝑢𝑧)
54 resabs1 4894 . . . . . . . . . 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 4500 . . . . . . . . . . . . 13 (Ord 𝑧 → ¬ 𝑧𝑧)
5751, 56syl 14 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑧𝑧)
58 fsnunres 5668 . . . . . . . . . . . 12 ((𝑔 Fn 𝑧 ∧ ¬ 𝑧𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) = 𝑔)
5918, 57, 58syl2anc 409 . . . . . . . . . . 11 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) = 𝑔)
6059reseq1d 4864 . . . . . . . . . 10 (𝜑 → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔𝑢))
6160adantr 274 . . . . . . . . 9 ((𝜑𝑢𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔𝑢))
6255, 61eqtr3d 2192 . . . . . . . 8 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢) = (𝑔𝑢))
6362fveq2d 5471 . . . . . . 7 ((𝜑𝑢𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)) = (𝐺‘(𝑔𝑢)))
6435, 49, 633eqtr4d 2200 . . . . . 6 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
65 fneq2 5258 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
6665imbi1d 230 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
6766albidv 1804 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
68133expia 1187 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
6968alrimiv 1854 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
7069ralrimiva 2530 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
7167, 70, 17rspcdva 2821 . . . . . . . . . 10 (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
72 fneq1 5257 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
73 fveq2 5467 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
7473eleq1d 2226 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
7572, 74imbi12d 233 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
7675spv 1840 . . . . . . . . . 10 (∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
7771, 18, 76sylc 62 . . . . . . . . 9 (𝜑 → (𝐺𝑔) ∈ V)
78 fndm 5268 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
7918, 78syl 14 . . . . . . . . . 10 (𝜑 → dom 𝑔 = 𝑧)
8057, 79neleqtrrd 2256 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
81 fsnunfv 5667 . . . . . . . . 9 ((𝑧𝑌 ∧ (𝐺𝑔) ∈ V ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧) = (𝐺𝑔))
825, 77, 80, 81syl3anc 1220 . . . . . . . 8 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧) = (𝐺𝑔))
8382adantr 274 . . . . . . 7 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧) = (𝐺𝑔))
84 simpr 109 . . . . . . . 8 ((𝜑𝑢 = 𝑧) → 𝑢 = 𝑧)
8584fveq2d 5471 . . . . . . 7 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧))
86 reseq2 4860 . . . . . . . . 9 (𝑢 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧))
8786, 59sylan9eqr 2212 . . . . . . . 8 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢) = 𝑔)
8887fveq2d 5471 . . . . . . 7 ((𝜑𝑢 = 𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)) = (𝐺𝑔))
8983, 85, 883eqtr4d 2200 . . . . . 6 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
9064, 89jaodan 787 . . . . 5 ((𝜑 ∧ (𝑢𝑧𝑢 = 𝑧)) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
9122, 90sylan2b 285 . . . 4 ((𝜑𝑢 ∈ suc 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
9291ralrimiva 2530 . . 3 (𝜑 → ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
93 fneq2 5258 . . . . 5 (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn 𝑤 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧))
94 raleq 2652 . . . . 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 465 . . . 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 2816 . . 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 1218 . 2 (𝜑 → ∃𝑤𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢))))
98 vex 2715 . . . . . 6 𝑧 ∈ V
99 opexg 4188 . . . . . 6 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
10098, 77, 99sylancr 411 . . . . 5 (𝜑 → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
101 snexg 4145 . . . . 5 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
102100, 101syl 14 . . . 4 (𝜑 → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
103 unexg 4402 . . . 4 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
10423, 102, 103sylancr 411 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
10514tfr1onlem3ag 6281 . . 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 166 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ 𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 963  ∀wal 1333   = wceq 1335   ∈ wcel 2128  {cab 2143   ≠ wne 2327  ∀wral 2435  ∃wrex 2436  Vcvv 2712   ∪ cun 3100   ⊆ wss 3102  {csn 3560  ⟨cop 3563  ∪ cuni 3772  Ord word 4322  Oncon0 4323  suc csuc 4325  dom cdm 4585   ↾ cres 4587  Fun wfun 5163   Fn wfn 5164  ‘cfv 5169  recscrecs 6248 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-res 4597  df-iota 5134  df-fun 5171  df-fn 5172  df-fv 5177 This theorem is referenced by:  tfr1onlembacc  6286  tfr1onlemres  6293
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