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Theorem tfrlemisucaccv 5880
Description: We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 5887. (Contributed by Jim Kingdon, 4-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfrlemisucfn.1 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
tfrlemisucfn.2 (φx(Fun 𝐹 (𝐹x) V))
tfrlemisucfn.3 (φz On)
tfrlemisucfn.4 (φg Fn z)
tfrlemisucfn.5 (φg A)
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 4193 . . . 4 (z On → suc z On)
31, 2syl 14 . . 3 (φ → suc z On)
4 tfrlemisucfn.1 . . . 4 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
5 tfrlemisucfn.2 . . . 4 (φx(Fun 𝐹 (𝐹x) V))
6 tfrlemisucfn.4 . . . 4 (φg Fn z)
7 tfrlemisucfn.5 . . . 4 (φg A)
84, 5, 1, 6, 7tfrlemisucfn 5879 . . 3 (φ → (g ∪ {⟨z, (𝐹g)⟩}) Fn suc z)
9 vex 2554 . . . . . 6 u V
109elsuc 4109 . . . . 5 (u suc z ↔ (u z u = z))
11 vex 2554 . . . . . . . . . . 11 g V
124, 11tfrlem3a 5866 . . . . . . . . . 10 (g Av On (g Fn v u v (gu) = (𝐹‘(gu))))
137, 12sylib 127 . . . . . . . . 9 (φv On (g Fn v u v (gu) = (𝐹‘(gu))))
14 simprrr 492 . . . . . . . . . 10 ((φ (v On (g Fn v u v (gu) = (𝐹‘(gu))))) → u v (gu) = (𝐹‘(gu)))
15 simprrl 491 . . . . . . . . . . . 12 ((φ (v On (g Fn v u v (gu) = (𝐹‘(gu))))) → g Fn v)
166adantr 261 . . . . . . . . . . . 12 ((φ (v On (g Fn v u v (gu) = (𝐹‘(gu))))) → g Fn z)
17 fndmu 4943 . . . . . . . . . . . 12 ((g Fn v g Fn z) → v = z)
1815, 16, 17syl2anc 391 . . . . . . . . . . 11 ((φ (v On (g Fn v u v (gu) = (𝐹‘(gu))))) → v = z)
1918raleqdv 2505 . . . . . . . . . 10 ((φ (v On (g Fn v u v (gu) = (𝐹‘(gu))))) → (u v (gu) = (𝐹‘(gu)) ↔ u z (gu) = (𝐹‘(gu))))
2014, 19mpbid 135 . . . . . . . . 9 ((φ (v On (g Fn v u v (gu) = (𝐹‘(gu))))) → u z (gu) = (𝐹‘(gu)))
2113, 20rexlimddv 2431 . . . . . . . 8 (φu z (gu) = (𝐹‘(gu)))
2221r19.21bi 2401 . . . . . . 7 ((φ u z) → (gu) = (𝐹‘(gu)))
23 elirrv 4226 . . . . . . . . . . 11 ¬ u u
24 elequ2 1598 . . . . . . . . . . 11 (z = u → (u zu u))
2523, 24mtbiri 599 . . . . . . . . . 10 (z = u → ¬ u z)
2625necon2ai 2253 . . . . . . . . 9 (u zzu)
2726adantl 262 . . . . . . . 8 ((φ u z) → zu)
28 fvunsng 5300 . . . . . . . 8 ((u V zu) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (gu))
299, 27, 28sylancr 393 . . . . . . 7 ((φ u z) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (gu))
30 eloni 4078 . . . . . . . . . . . 12 (z On → Ord z)
311, 30syl 14 . . . . . . . . . . 11 (φ → Ord z)
32 ordelss 4082 . . . . . . . . . . 11 ((Ord z u z) → uz)
3331, 32sylan 267 . . . . . . . . . 10 ((φ u z) → uz)
34 resabs1 4583 . . . . . . . . . 10 (uz → (((g ∪ {⟨z, (𝐹g)⟩}) ↾ z) ↾ u) = ((g ∪ {⟨z, (𝐹g)⟩}) ↾ u))
3533, 34syl 14 . . . . . . . . 9 ((φ u z) → (((g ∪ {⟨z, (𝐹g)⟩}) ↾ z) ↾ u) = ((g ∪ {⟨z, (𝐹g)⟩}) ↾ u))
36 elirrv 4226 . . . . . . . . . . . 12 ¬ z z
37 fsnunres 5307 . . . . . . . . . . . 12 ((g Fn z ¬ z z) → ((g ∪ {⟨z, (𝐹g)⟩}) ↾ z) = g)
386, 36, 37sylancl 392 . . . . . . . . . . 11 (φ → ((g ∪ {⟨z, (𝐹g)⟩}) ↾ z) = g)
3938reseq1d 4554 . . . . . . . . . 10 (φ → (((g ∪ {⟨z, (𝐹g)⟩}) ↾ z) ↾ u) = (gu))
4039adantr 261 . . . . . . . . 9 ((φ u z) → (((g ∪ {⟨z, (𝐹g)⟩}) ↾ z) ↾ u) = (gu))
4135, 40eqtr3d 2071 . . . . . . . 8 ((φ u z) → ((g ∪ {⟨z, (𝐹g)⟩}) ↾ u) = (gu))
4241fveq2d 5125 . . . . . . 7 ((φ u z) → (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)) = (𝐹‘(gu)))
4322, 29, 423eqtr4d 2079 . . . . . 6 ((φ u z) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)))
445tfrlem3-2d 5869 . . . . . . . . . 10 (φ → (Fun 𝐹 (𝐹g) V))
4544simprd 107 . . . . . . . . 9 (φ → (𝐹g) V)
46 fndm 4941 . . . . . . . . . . . 12 (g Fn z → dom g = z)
476, 46syl 14 . . . . . . . . . . 11 (φ → dom g = z)
4847eleq2d 2104 . . . . . . . . . 10 (φ → (z dom gz z))
4936, 48mtbiri 599 . . . . . . . . 9 (φ → ¬ z dom g)
50 fsnunfv 5306 . . . . . . . . 9 ((z On (𝐹g) V ¬ z dom g) → ((g ∪ {⟨z, (𝐹g)⟩})‘z) = (𝐹g))
511, 45, 49, 50syl3anc 1134 . . . . . . . 8 (φ → ((g ∪ {⟨z, (𝐹g)⟩})‘z) = (𝐹g))
5251adantr 261 . . . . . . 7 ((φ u = z) → ((g ∪ {⟨z, (𝐹g)⟩})‘z) = (𝐹g))
53 simpr 103 . . . . . . . 8 ((φ u = z) → u = z)
5453fveq2d 5125 . . . . . . 7 ((φ u = z) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = ((g ∪ {⟨z, (𝐹g)⟩})‘z))
55 reseq2 4550 . . . . . . . . 9 (u = z → ((g ∪ {⟨z, (𝐹g)⟩}) ↾ u) = ((g ∪ {⟨z, (𝐹g)⟩}) ↾ z))
5655, 38sylan9eqr 2091 . . . . . . . 8 ((φ u = z) → ((g ∪ {⟨z, (𝐹g)⟩}) ↾ u) = g)
5756fveq2d 5125 . . . . . . 7 ((φ u = z) → (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)) = (𝐹g))
5852, 54, 573eqtr4d 2079 . . . . . 6 ((φ u = z) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)))
5943, 58jaodan 709 . . . . 5 ((φ (u z u = z)) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)))
6010, 59sylan2b 271 . . . 4 ((φ u suc z) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)))
6160ralrimiva 2386 . . 3 (φu suc z((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)))
62 fneq2 4931 . . . . 5 (w = suc z → ((g ∪ {⟨z, (𝐹g)⟩}) Fn w ↔ (g ∪ {⟨z, (𝐹g)⟩}) Fn suc z))
63 raleq 2499 . . . . 5 (w = suc z → (u w ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)) ↔ u suc z((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u))))
6462, 63anbi12d 442 . . . 4 (w = suc z → (((g ∪ {⟨z, (𝐹g)⟩}) Fn w u w ((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 2650 . . 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 u w ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u))))
663, 8, 61, 65syl12anc 1132 . 2 (φw On ((g ∪ {⟨z, (𝐹g)⟩}) Fn w u w ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u))))
67 vex 2554 . . . . . 6 z V
68 opexg 3955 . . . . . 6 ((z V (𝐹g) V) → ⟨z, (𝐹g)⟩ V)
6967, 45, 68sylancr 393 . . . . 5 (φ → ⟨z, (𝐹g)⟩ V)
70 snexg 3927 . . . . 5 (⟨z, (𝐹g)⟩ V → {⟨z, (𝐹g)⟩} V)
7169, 70syl 14 . . . 4 (φ → {⟨z, (𝐹g)⟩} V)
72 unexg 4144 . . . 4 ((g V {⟨z, (𝐹g)⟩} V) → (g ∪ {⟨z, (𝐹g)⟩}) V)
7311, 71, 72sylancr 393 . . 3 (φ → (g ∪ {⟨z, (𝐹g)⟩}) V)
744tfrlem3ag 5865 . . 3 ((g ∪ {⟨z, (𝐹g)⟩}) V → ((g ∪ {⟨z, (𝐹g)⟩}) Aw On ((g ∪ {⟨z, (𝐹g)⟩}) Fn w u w ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)))))
7573, 74syl 14 . 2 (φ → ((g ∪ {⟨z, (𝐹g)⟩}) Aw On ((g ∪ {⟨z, (𝐹g)⟩}) Fn w u w ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)))))
7666, 75mpbird 156 1 (φ → (g ∪ {⟨z, (𝐹g)⟩}) A)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   wa 97  wb 98   wo 628  wal 1240   = wceq 1242   wcel 1390  {cab 2023  wne 2201  wral 2300  wrex 2301  Vcvv 2551  cun 2909  wss 2911  {csn 3367  cop 3370  Ord word 4065  Oncon0 4066  suc csuc 4068  dom cdm 4288  cres 4290  Fun wfun 4839   Fn wfn 4840  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  tfrlemibacc  5881  tfrlemi14d  5888
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