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Theorem tfrlemisucaccv 5969
 Description: We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 5976. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
tfrlemisucfn.2 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
tfrlemisucfn.3 (𝜑𝑧 ∈ On)
tfrlemisucfn.4 (𝜑𝑔 Fn 𝑧)
tfrlemisucfn.5 (𝜑𝑔𝐴)
Assertion
Ref Expression
tfrlemisucaccv (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴)
Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝑧,𝐴   𝑓,𝐹,𝑔,𝑥,𝑦,𝑧   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑧,𝑓,𝑔)

Proof of Theorem tfrlemisucaccv
Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrlemisucfn.3 . . . 4 (𝜑𝑧 ∈ On)
2 suceloni 4254 . . . 4 (𝑧 ∈ On → suc 𝑧 ∈ On)
31, 2syl 14 . . 3 (𝜑 → suc 𝑧 ∈ On)
4 tfrlemisucfn.1 . . . 4 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐹‘(𝑓𝑦)))}
5 tfrlemisucfn.2 . . . 4 (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹𝑥) ∈ V))
6 tfrlemisucfn.4 . . . 4 (𝜑𝑔 Fn 𝑧)
7 tfrlemisucfn.5 . . . 4 (𝜑𝑔𝐴)
84, 5, 1, 6, 7tfrlemisucfn 5968 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
9 vex 2577 . . . . . 6 𝑢 ∈ V
109elsuc 4170 . . . . 5 (𝑢 ∈ suc 𝑧 ↔ (𝑢𝑧𝑢 = 𝑧))
11 vex 2577 . . . . . . . . . . 11 𝑔 ∈ V
124, 11tfrlem3a 5955 . . . . . . . . . 10 (𝑔𝐴 ↔ ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
137, 12sylib 131 . . . . . . . . 9 (𝜑 → ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
14 simprrr 500 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))
15 simprrl 499 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → 𝑔 Fn 𝑣)
166adantr 265 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → 𝑔 Fn 𝑧)
17 fndmu 5027 . . . . . . . . . . . 12 ((𝑔 Fn 𝑣𝑔 Fn 𝑧) → 𝑣 = 𝑧)
1815, 16, 17syl2anc 397 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → 𝑣 = 𝑧)
1918raleqdv 2528 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → (∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
2014, 19mpbid 139 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))
2113, 20rexlimddv 2454 . . . . . . . 8 (𝜑 → ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))
2221r19.21bi 2424 . . . . . . 7 ((𝜑𝑢𝑧) → (𝑔𝑢) = (𝐹‘(𝑔𝑢)))
23 elirrv 4299 . . . . . . . . . . 11 ¬ 𝑢𝑢
24 elequ2 1617 . . . . . . . . . . 11 (𝑧 = 𝑢 → (𝑢𝑧𝑢𝑢))
2523, 24mtbiri 610 . . . . . . . . . 10 (𝑧 = 𝑢 → ¬ 𝑢𝑧)
2625necon2ai 2274 . . . . . . . . 9 (𝑢𝑧𝑧𝑢)
2726adantl 266 . . . . . . . 8 ((𝜑𝑢𝑧) → 𝑧𝑢)
28 fvunsng 5384 . . . . . . . 8 ((𝑢 ∈ V ∧ 𝑧𝑢) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝑔𝑢))
299, 27, 28sylancr 399 . . . . . . 7 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝑔𝑢))
30 eloni 4139 . . . . . . . . . . . 12 (𝑧 ∈ On → Ord 𝑧)
311, 30syl 14 . . . . . . . . . . 11 (𝜑 → Ord 𝑧)
32 ordelss 4143 . . . . . . . . . . 11 ((Ord 𝑧𝑢𝑧) → 𝑢𝑧)
3331, 32sylan 271 . . . . . . . . . 10 ((𝜑𝑢𝑧) → 𝑢𝑧)
34 resabs1 4667 . . . . . . . . . 10 (𝑢𝑧 → (((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢))
3533, 34syl 14 . . . . . . . . 9 ((𝜑𝑢𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢))
36 elirrv 4299 . . . . . . . . . . . 12 ¬ 𝑧𝑧
37 fsnunres 5391 . . . . . . . . . . . 12 ((𝑔 Fn 𝑧 ∧ ¬ 𝑧𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧) = 𝑔)
386, 36, 37sylancl 398 . . . . . . . . . . 11 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧) = 𝑔)
3938reseq1d 4638 . . . . . . . . . 10 (𝜑 → (((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔𝑢))
4039adantr 265 . . . . . . . . 9 ((𝜑𝑢𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔𝑢))
4135, 40eqtr3d 2090 . . . . . . . 8 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢) = (𝑔𝑢))
4241fveq2d 5209 . . . . . . 7 ((𝜑𝑢𝑧) → (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)) = (𝐹‘(𝑔𝑢)))
4322, 29, 423eqtr4d 2098 . . . . . 6 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))
445tfrlem3-2d 5958 . . . . . . . . . 10 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
4544simprd 111 . . . . . . . . 9 (𝜑 → (𝐹𝑔) ∈ V)
46 fndm 5025 . . . . . . . . . . . 12 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
476, 46syl 14 . . . . . . . . . . 11 (𝜑 → dom 𝑔 = 𝑧)
4847eleq2d 2123 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ dom 𝑔𝑧𝑧))
4936, 48mtbiri 610 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
50 fsnunfv 5390 . . . . . . . . 9 ((𝑧 ∈ On ∧ (𝐹𝑔) ∈ V ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑧) = (𝐹𝑔))
511, 45, 49, 50syl3anc 1146 . . . . . . . 8 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑧) = (𝐹𝑔))
5251adantr 265 . . . . . . 7 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑧) = (𝐹𝑔))
53 simpr 107 . . . . . . . 8 ((𝜑𝑢 = 𝑧) → 𝑢 = 𝑧)
5453fveq2d 5209 . . . . . . 7 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑧))
55 reseq2 4634 . . . . . . . . 9 (𝑢 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧))
5655, 38sylan9eqr 2110 . . . . . . . 8 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢) = 𝑔)
5756fveq2d 5209 . . . . . . 7 ((𝜑𝑢 = 𝑧) → (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)) = (𝐹𝑔))
5852, 54, 573eqtr4d 2098 . . . . . 6 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))
5943, 58jaodan 721 . . . . 5 ((𝜑 ∧ (𝑢𝑧𝑢 = 𝑧)) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))
6010, 59sylan2b 275 . . . 4 ((𝜑𝑢 ∈ suc 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))
6160ralrimiva 2409 . . 3 (𝜑 → ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))
62 fneq2 5015 . . . . 5 (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn 𝑤 ↔ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧))
63 raleq 2522 . . . . 5 (𝑤 = suc 𝑧 → (∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)) ↔ ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢))))
6462, 63anbi12d 450 . . . 4 (𝑤 = suc 𝑧 → (((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢))) ↔ ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧 ∧ ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))))
6564rspcev 2673 . . 3 ((suc 𝑧 ∈ On ∧ ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧 ∧ ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))) → ∃𝑤 ∈ On ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢))))
663, 8, 61, 65syl12anc 1144 . 2 (𝜑 → ∃𝑤 ∈ On ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢))))
67 vex 2577 . . . . . 6 𝑧 ∈ V
68 opexg 3991 . . . . . 6 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
6967, 45, 68sylancr 399 . . . . 5 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
70 snexg 3963 . . . . 5 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
7169, 70syl 14 . . . 4 (𝜑 → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
72 unexg 4205 . . . 4 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
7311, 71, 72sylancr 399 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
744tfrlem3ag 5954 . . 3 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴 ↔ ∃𝑤 ∈ On ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))))
7573, 74syl 14 . 2 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴 ↔ ∃𝑤 ∈ On ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))))
7666, 75mpbird 160 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ↔ wb 102   ∨ wo 639  ∀wal 1257   = wceq 1259   ∈ wcel 1409  {cab 2042   ≠ wne 2220  ∀wral 2323  ∃wrex 2324  Vcvv 2574   ∪ cun 2942   ⊆ wss 2944  {csn 3402  ⟨cop 3405  Ord word 4126  Oncon0 4127  suc csuc 4129  dom cdm 4372   ↾ cres 4374  Fun wfun 4923   Fn wfn 4924  ‘cfv 4929 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-tr 3882  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-res 4384  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937 This theorem is referenced by:  tfrlemibacc  5970  tfrlemi14d  5977
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