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Theorem tfrlemisucaccv 6326
Description: We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 6333. (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 onsuc 4501 . . . 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 6325 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧)
9 vex 2741 . . . . . 6 𝑢 ∈ V
109elsuc 4407 . . . . 5 (𝑢 ∈ suc 𝑧 ↔ (𝑢𝑧𝑢 = 𝑧))
11 vex 2741 . . . . . . . . . . 11 𝑔 ∈ V
124, 11tfrlem3a 6311 . . . . . . . . . 10 (𝑔𝐴 ↔ ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
137, 12sylib 122 . . . . . . . . 9 (𝜑 → ∃𝑣 ∈ On (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
14 simprrr 540 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))
15 simprrl 539 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → 𝑔 Fn 𝑣)
166adantr 276 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → 𝑔 Fn 𝑧)
17 fndmu 5318 . . . . . . . . . . . 12 ((𝑔 Fn 𝑣𝑔 Fn 𝑧) → 𝑣 = 𝑧)
1815, 16, 17syl2anc 411 . . . . . . . . . . 11 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → 𝑣 = 𝑧)
1918raleqdv 2679 . . . . . . . . . 10 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → (∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢)) ↔ ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))
2014, 19mpbid 147 . . . . . . . . 9 ((𝜑 ∧ (𝑣 ∈ On ∧ (𝑔 Fn 𝑣 ∧ ∀𝑢𝑣 (𝑔𝑢) = (𝐹‘(𝑔𝑢))))) → ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))
2113, 20rexlimddv 2599 . . . . . . . 8 (𝜑 → ∀𝑢𝑧 (𝑔𝑢) = (𝐹‘(𝑔𝑢)))
2221r19.21bi 2565 . . . . . . 7 ((𝜑𝑢𝑧) → (𝑔𝑢) = (𝐹‘(𝑔𝑢)))
23 elirrv 4548 . . . . . . . . . . 11 ¬ 𝑢𝑢
24 elequ2 2153 . . . . . . . . . . 11 (𝑧 = 𝑢 → (𝑢𝑧𝑢𝑢))
2523, 24mtbiri 675 . . . . . . . . . 10 (𝑧 = 𝑢 → ¬ 𝑢𝑧)
2625necon2ai 2401 . . . . . . . . 9 (𝑢𝑧𝑧𝑢)
2726adantl 277 . . . . . . . 8 ((𝜑𝑢𝑧) → 𝑧𝑢)
28 fvunsng 5711 . . . . . . . 8 ((𝑢 ∈ V ∧ 𝑧𝑢) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝑔𝑢))
299, 27, 28sylancr 414 . . . . . . 7 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝑔𝑢))
30 eloni 4376 . . . . . . . . . . . 12 (𝑧 ∈ On → Ord 𝑧)
311, 30syl 14 . . . . . . . . . . 11 (𝜑 → Ord 𝑧)
32 ordelss 4380 . . . . . . . . . . 11 ((Ord 𝑧𝑢𝑧) → 𝑢𝑧)
3331, 32sylan 283 . . . . . . . . . 10 ((𝜑𝑢𝑧) → 𝑢𝑧)
34 resabs1 4937 . . . . . . . . . 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 4548 . . . . . . . . . . . 12 ¬ 𝑧𝑧
37 fsnunres 5719 . . . . . . . . . . . 12 ((𝑔 Fn 𝑧 ∧ ¬ 𝑧𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧) = 𝑔)
386, 36, 37sylancl 413 . . . . . . . . . . 11 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧) = 𝑔)
3938reseq1d 4907 . . . . . . . . . 10 (𝜑 → (((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔𝑢))
4039adantr 276 . . . . . . . . 9 ((𝜑𝑢𝑧) → (((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧) ↾ 𝑢) = (𝑔𝑢))
4135, 40eqtr3d 2212 . . . . . . . 8 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢) = (𝑔𝑢))
4241fveq2d 5520 . . . . . . 7 ((𝜑𝑢𝑧) → (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)) = (𝐹‘(𝑔𝑢)))
4322, 29, 423eqtr4d 2220 . . . . . 6 ((𝜑𝑢𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))
445tfrlem3-2d 6313 . . . . . . . . . 10 (𝜑 → (Fun 𝐹 ∧ (𝐹𝑔) ∈ V))
4544simprd 114 . . . . . . . . 9 (𝜑 → (𝐹𝑔) ∈ V)
46 fndm 5316 . . . . . . . . . . . 12 (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
476, 46syl 14 . . . . . . . . . . 11 (𝜑 → dom 𝑔 = 𝑧)
4847eleq2d 2247 . . . . . . . . . 10 (𝜑 → (𝑧 ∈ dom 𝑔𝑧𝑧))
4936, 48mtbiri 675 . . . . . . . . 9 (𝜑 → ¬ 𝑧 ∈ dom 𝑔)
50 fsnunfv 5718 . . . . . . . . 9 ((𝑧 ∈ On ∧ (𝐹𝑔) ∈ V ∧ ¬ 𝑧 ∈ dom 𝑔) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑧) = (𝐹𝑔))
511, 45, 49, 50syl3anc 1238 . . . . . . . 8 (𝜑 → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑧) = (𝐹𝑔))
5251adantr 276 . . . . . . 7 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑧) = (𝐹𝑔))
53 simpr 110 . . . . . . . 8 ((𝜑𝑢 = 𝑧) → 𝑢 = 𝑧)
5453fveq2d 5520 . . . . . . 7 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑧))
55 reseq2 4903 . . . . . . . . 9 (𝑢 = 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢) = ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑧))
5655, 38sylan9eqr 2232 . . . . . . . 8 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢) = 𝑔)
5756fveq2d 5520 . . . . . . 7 ((𝜑𝑢 = 𝑧) → (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)) = (𝐹𝑔))
5852, 54, 573eqtr4d 2220 . . . . . 6 ((𝜑𝑢 = 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))
5943, 58jaodan 797 . . . . 5 ((𝜑 ∧ (𝑢𝑧𝑢 = 𝑧)) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))
6010, 59sylan2b 287 . . . 4 ((𝜑𝑢 ∈ suc 𝑧) → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))
6160ralrimiva 2550 . . 3 (𝜑 → ∀𝑢 ∈ suc 𝑧((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢)))
62 fneq2 5306 . . . . 5 (𝑤 = suc 𝑧 → ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn 𝑤 ↔ (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn suc 𝑧))
63 raleq 2673 . . . . 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 473 . . . 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 2842 . . 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 1236 . 2 (𝜑 → ∃𝑤 ∈ On ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) Fn 𝑤 ∧ ∀𝑢𝑤 ((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩})‘𝑢) = (𝐹‘((𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ↾ 𝑢))))
67 vex 2741 . . . . . 6 𝑧 ∈ V
68 opexg 4229 . . . . . 6 ((𝑧 ∈ V ∧ (𝐹𝑔) ∈ V) → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
6967, 45, 68sylancr 414 . . . . 5 (𝜑 → ⟨𝑧, (𝐹𝑔)⟩ ∈ V)
70 snexg 4185 . . . . 5 (⟨𝑧, (𝐹𝑔)⟩ ∈ V → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
7169, 70syl 14 . . . 4 (𝜑 → {⟨𝑧, (𝐹𝑔)⟩} ∈ V)
72 unexg 4444 . . . 4 ((𝑔 ∈ V ∧ {⟨𝑧, (𝐹𝑔)⟩} ∈ V) → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
7311, 71, 72sylancr 414 . . 3 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ V)
744tfrlem3ag 6310 . . 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 167 1 (𝜑 → (𝑔 ∪ {⟨𝑧, (𝐹𝑔)⟩}) ∈ 𝐴)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  wal 1351   = wceq 1353  wcel 2148  {cab 2163  wne 2347  wral 2455  wrex 2456  Vcvv 2738  cun 3128  wss 3130  {csn 3593  cop 3596  Ord word 4363  Oncon0 4364  suc csuc 4366  dom cdm 4627  cres 4629  Fun wfun 5211   Fn wfn 5212  cfv 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-res 4639  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225
This theorem is referenced by:  tfrlemibacc  6327  tfrlemi14d  6334
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