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Theorem tfr1onlemsucaccv 6010
Description: Lemma for tfr1on 6019. 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 4185 . . . . 5 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
21eleq1d 2151 . . . 4 (𝑥 = 𝑧 → (suc 𝑥𝑋 ↔ suc 𝑧𝑋))
3 tfr1onlemsucaccv.u . . . . 5 ((𝜑𝑥 𝑋) → suc 𝑥𝑋)
43ralrimiva 2439 . . . 4 (𝜑 → ∀𝑥 𝑋 suc 𝑥𝑋)
5 tfr1onlemsucaccv.zy . . . . 5 (𝜑𝑧𝑌)
6 tfr1onlemsucaccv.yx . . . . 5 (𝜑𝑌𝑋)
7 elunii 3626 . . . . 5 ((𝑧𝑌𝑌𝑋) → 𝑧 𝑋)
85, 6, 7syl2anc 403 . . . 4 (𝜑𝑧 𝑋)
92, 4, 8rspcdva 2715 . . 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 4171 . . . . 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 6009 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧)
21 vex 2613 . . . . . 6 𝑢 ∈ V
2221elsuc 4189 . . . . 5 (𝑢 ∈ suc 𝑧 ↔ (𝑢𝑧𝑢 = 𝑧))
23 vex 2613 . . . . . . . . . . 11 𝑔 ∈ V
2414tfr1onlem3ag 6006 . . . . . . . . . . 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 5051 . . . . . . . . . . . 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 2560 . . . . . . . . . 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 2486 . . . . . . . 8 (𝜑 → ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))
3534r19.21bi 2454 . . . . . . 7 ((𝜑𝑢𝑧) → (𝑔𝑢) = (𝐺‘(𝑔𝑢)))
36 ordelon 4166 . . . . . . . . . . . . 13 ((Ord 𝑋𝑧𝑋) → 𝑧 ∈ On)
3712, 17, 36syl2anc 403 . . . . . . . . . . . 12 (𝜑𝑧 ∈ On)
38 onelon 4167 . . . . . . . . . . . 12 ((𝑧 ∈ On ∧ 𝑢𝑧) → 𝑢 ∈ On)
3937, 38sylan 277 . . . . . . . . . . 11 ((𝜑𝑢𝑧) → 𝑢 ∈ On)
40 eloni 4158 . . . . . . . . . . 11 (𝑢 ∈ On → Ord 𝑢)
41 ordirr 4313 . . . . . . . . . . 11 (Ord 𝑢 → ¬ 𝑢𝑢)
4239, 40, 413syl 17 . . . . . . . . . 10 ((𝜑𝑢𝑧) → ¬ 𝑢𝑢)
43 elequ2 1643 . . . . . . . . . . . 12 (𝑧 = 𝑢 → (𝑢𝑧𝑢𝑢))
4443biimpcd 157 . . . . . . . . . . 11 (𝑢𝑧 → (𝑧 = 𝑢𝑢𝑢))
4544adantl 271 . . . . . . . . . 10 ((𝜑𝑢𝑧) → (𝑧 = 𝑢𝑢𝑢))
4642, 45mtod 622 . . . . . . . . 9 ((𝜑𝑢𝑧) → ¬ 𝑧 = 𝑢)
4746neqned 2256 . . . . . . . 8 ((𝜑𝑢𝑧) → 𝑧𝑢)
48 fvunsng 5409 . . . . . . . 8 ((𝑢 ∈ V ∧ 𝑧𝑢) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝑔𝑢))
4921, 47, 48sylancr 405 . . . . . . 7 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝑔𝑢))
50 eloni 4158 . . . . . . . . . . . 12 (𝑧 ∈ On → Ord 𝑧)
5137, 50syl 14 . . . . . . . . . . 11 (𝜑 → Ord 𝑧)
52 ordelss 4162 . . . . . . . . . . 11 ((Ord 𝑧𝑢𝑧) → 𝑢𝑧)
5351, 52sylan 277 . . . . . . . . . 10 ((𝜑𝑢𝑧) → 𝑢𝑧)
54 resabs1 4688 . . . . . . . . . 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 4313 . . . . . . . . . . . . 13 (Ord 𝑧 → ¬ 𝑧𝑧)
5751, 56syl 14 . . . . . . . . . . . 12 (𝜑 → ¬ 𝑧𝑧)
58 fsnunres 5416 . . . . . . . . . . . 12 ((𝑔 Fn 𝑧 ∧ ¬ 𝑧𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) = 𝑔)
5918, 57, 58syl2anc 403 . . . . . . . . . . 11 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) = 𝑔)
6059reseq1d 4659 . . . . . . . . . 10 (𝜑 → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔𝑢))
6160adantr 270 . . . . . . . . 9 ((𝜑𝑢𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔𝑢))
6255, 61eqtr3d 2117 . . . . . . . 8 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢) = (𝑔𝑢))
6362fveq2d 5233 . . . . . . 7 ((𝜑𝑢𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)) = (𝐺‘(𝑔𝑢)))
6435, 49, 633eqtr4d 2125 . . . . . 6 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
65 fneq2 5039 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑓 Fn 𝑥𝑓 Fn 𝑧))
6665imbi1d 229 . . . . . . . . . . . 12 (𝑥 = 𝑧 → ((𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ (𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
6766albidv 1747 . . . . . . . . . . 11 (𝑥 = 𝑧 → (∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V) ↔ ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V)))
68133expia 1141 . . . . . . . . . . . . 13 ((𝜑𝑥𝑋) → (𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
6968alrimiv 1797 . . . . . . . . . . . 12 ((𝜑𝑥𝑋) → ∀𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
7069ralrimiva 2439 . . . . . . . . . . 11 (𝜑 → ∀𝑥𝑋𝑓(𝑓 Fn 𝑥 → (𝐺𝑓) ∈ V))
7167, 70, 17rspcdva 2715 . . . . . . . . . 10 (𝜑 → ∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V))
72 fneq1 5038 . . . . . . . . . . . 12 (𝑓 = 𝑔 → (𝑓 Fn 𝑧𝑔 Fn 𝑧))
73 fveq2 5229 . . . . . . . . . . . . 13 (𝑓 = 𝑔 → (𝐺𝑓) = (𝐺𝑔))
7473eleq1d 2151 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ((𝐺𝑓) ∈ V ↔ (𝐺𝑔) ∈ V))
7572, 74imbi12d 232 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) ↔ (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V)))
7675spv 1783 . . . . . . . . . 10 (∀𝑓(𝑓 Fn 𝑧 → (𝐺𝑓) ∈ V) → (𝑔 Fn 𝑧 → (𝐺𝑔) ∈ V))
7771, 18, 76sylc 61 . . . . . . . . 9 (𝜑 → (𝐺𝑔) ∈ V)
78 fndm 5049 . . . . . . . . . . 11 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
7918, 78syl 14 . . . . . . . . . 10 (𝜑 → dom 𝑔 = 𝑧)
8057, 79neleqtrrd 2181 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
81 fsnunfv 5415 . . . . . . . . 9 ((𝑧𝑌 ∧ (𝐺𝑔) ∈ V ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧) = (𝐺𝑔))
825, 77, 80, 81syl3anc 1170 . . . . . . . 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 5233 . . . . . . 7 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑧))
86 reseq2 4655 . . . . . . . . 9 (𝑢 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑧))
8786, 59sylan9eqr 2137 . . . . . . . 8 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢) = 𝑔)
8887fveq2d 5233 . . . . . . 7 ((𝜑𝑢 = 𝑧) → (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)) = (𝐺𝑔))
8983, 85, 883eqtr4d 2125 . . . . . 6 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
9064, 89jaodan 744 . . . . 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 2439 . . 3 (𝜑 → ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢)))
93 fneq2 5039 . . . . 5 (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn 𝑤 ↔ (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn suc 𝑧))
94 raleq 2554 . . . . 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 2710 . . 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 1168 . 2 (𝜑 → ∃𝑤𝑋 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩})‘𝑢) = (𝐺‘((𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ↾ 𝑢))))
98 vex 2613 . . . . . 6 𝑧 ∈ V
99 opexg 4011 . . . . . 6 ((𝑧 ∈ V ∧ (𝐺𝑔) ∈ V) → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
10098, 77, 99sylancr 405 . . . . 5 (𝜑 → ⟨𝑧, (𝐺𝑔)⟩ ∈ V)
101 snexg 3976 . . . . 5 (⟨𝑧, (𝐺𝑔)⟩ ∈ V → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
102100, 101syl 14 . . . 4 (𝜑 → {⟨𝑧, (𝐺𝑔)⟩} ∈ V)
103 unexg 4224 . . . 4 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐺𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
10423, 102, 103sylancr 405 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐺𝑔)⟩}) ∈ V)
10514tfr1onlem3ag 6006 . . 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 662  w3a 920  wal 1283   = wceq 1285  wcel 1434  {cab 2069  wne 2249  wral 2353  wrex 2354  Vcvv 2610  cun 2980  wss 2982  {csn 3416  cop 3419   cuni 3621  Ord word 4145  Oncon0 4146  suc csuc 4148  dom cdm 4391  cres 4393  Fun wfun 4946   Fn wfn 4947  cfv 4952  recscrecs 5973
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960
This theorem is referenced by:  tfr1onlembacc  6011  tfr1onlemres  6018
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