Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfrlemisucaccv Structured version   GIF version

Theorem tfrlemisucaccv 5860
 Description: We can extend an acceptable function by one element to produce an acceptable function. Lemma for tfrlemi1 5867. (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 4177 . . . 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 5859 . . 3 (φ → (g ∪ {⟨z, (𝐹g)⟩}) Fn suc z)
9 vex 2538 . . . . . 6 u V
109elsuc 4092 . . . . 5 (u suc z ↔ (u z u = z))
11 vex 2538 . . . . . . . . . . 11 g V
124, 11tfrlem3a 5847 . . . . . . . . . 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 480 . . . . . . . . . 10 ((φ (v On (g Fn v u v (gu) = (𝐹‘(gu))))) → u v (gu) = (𝐹‘(gu)))
15 simprrl 479 . . . . . . . . . . . 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 4926 . . . . . . . . . . . 12 ((g Fn v g Fn z) → v = z)
1815, 16, 17syl2anc 393 . . . . . . . . . . 11 ((φ (v On (g Fn v u v (gu) = (𝐹‘(gu))))) → v = z)
1918raleqdv 2489 . . . . . . . . . 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 2415 . . . . . . . 8 (φu z (gu) = (𝐹‘(gu)))
2221r19.21bi 2385 . . . . . . 7 ((φ u z) → (gu) = (𝐹‘(gu)))
23 elirrv 4210 . . . . . . . . . . 11 ¬ u u
24 elequ2 1583 . . . . . . . . . . 11 (z = u → (u zu u))
2523, 24mtbiri 587 . . . . . . . . . 10 (z = u → ¬ u z)
2625necon2ai 2237 . . . . . . . . 9 (u zzu)
2726adantl 262 . . . . . . . 8 ((φ u z) → zu)
28 fvunsng 5282 . . . . . . . 8 ((u V zu) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (gu))
299, 27, 28sylancr 395 . . . . . . 7 ((φ u z) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (gu))
30 eloni 4061 . . . . . . . . . . . 12 (z On → Ord z)
311, 30syl 14 . . . . . . . . . . 11 (φ → Ord z)
32 ordelss 4065 . . . . . . . . . . 11 ((Ord z u z) → uz)
3331, 32sylan 267 . . . . . . . . . 10 ((φ u z) → uz)
34 resabs1 4567 . . . . . . . . . 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 4210 . . . . . . . . . . . 12 ¬ z z
37 fsnunres 5289 . . . . . . . . . . . 12 ((g Fn z ¬ z z) → ((g ∪ {⟨z, (𝐹g)⟩}) ↾ z) = g)
386, 36, 37sylancl 394 . . . . . . . . . . 11 (φ → ((g ∪ {⟨z, (𝐹g)⟩}) ↾ z) = g)
3938reseq1d 4538 . . . . . . . . . 10 (φ → (((g ∪ {⟨z, (𝐹g)⟩}) ↾ z) ↾ u) = (gu))
4039adantr 261 . . . . . . . . 9 ((φ u z) → (((g ∪ {⟨z, (𝐹g)⟩}) ↾ z) ↾ u) = (gu))
4135, 40eqtr3d 2056 . . . . . . . 8 ((φ u z) → ((g ∪ {⟨z, (𝐹g)⟩}) ↾ u) = (gu))
4241fveq2d 5107 . . . . . . 7 ((φ u z) → (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)) = (𝐹‘(gu)))
4322, 29, 423eqtr4d 2064 . . . . . 6 ((φ u z) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)))
445tfrlem3-2d 5850 . . . . . . . . . 10 (φ → (Fun 𝐹 (𝐹g) V))
4544simprd 107 . . . . . . . . 9 (φ → (𝐹g) V)
46 fndm 4924 . . . . . . . . . . . 12 (g Fn z → dom g = z)
476, 46syl 14 . . . . . . . . . . 11 (φ → dom g = z)
4847eleq2d 2089 . . . . . . . . . 10 (φ → (z dom gz z))
4936, 48mtbiri 587 . . . . . . . . 9 (φ → ¬ z dom g)
50 fsnunfv 5288 . . . . . . . . 9 ((z On (𝐹g) V ¬ z dom g) → ((g ∪ {⟨z, (𝐹g)⟩})‘z) = (𝐹g))
511, 45, 49, 50syl3anc 1121 . . . . . . . 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 5107 . . . . . . 7 ((φ u = z) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = ((g ∪ {⟨z, (𝐹g)⟩})‘z))
55 reseq2 4534 . . . . . . . . 9 (u = z → ((g ∪ {⟨z, (𝐹g)⟩}) ↾ u) = ((g ∪ {⟨z, (𝐹g)⟩}) ↾ z))
5655, 38sylan9eqr 2076 . . . . . . . 8 ((φ u = z) → ((g ∪ {⟨z, (𝐹g)⟩}) ↾ u) = g)
5756fveq2d 5107 . . . . . . 7 ((φ u = z) → (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)) = (𝐹g))
5852, 54, 573eqtr4d 2064 . . . . . 6 ((φ u = z) → ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)))
5943, 58jaodan 697 . . . . 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 2370 . . 3 (φu suc z((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u)))
62 fneq2 4914 . . . . 5 (w = suc z → ((g ∪ {⟨z, (𝐹g)⟩}) Fn w ↔ (g ∪ {⟨z, (𝐹g)⟩}) Fn suc z))
63 raleq 2483 . . . . 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 445 . . . 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 2633 . . 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 1119 . 2 (φw On ((g ∪ {⟨z, (𝐹g)⟩}) Fn w u w ((g ∪ {⟨z, (𝐹g)⟩})‘u) = (𝐹‘((g ∪ {⟨z, (𝐹g)⟩}) ↾ u))))
67 vex 2538 . . . . . 6 z V
68 opexg 3938 . . . . . 6 ((z V (𝐹g) V) → ⟨z, (𝐹g)⟩ V)
6967, 45, 68sylancr 395 . . . . 5 (φ → ⟨z, (𝐹g)⟩ V)
70 snexg 3910 . . . . 5 (⟨z, (𝐹g)⟩ V → {⟨z, (𝐹g)⟩} V)
7169, 70syl 14 . . . 4 (φ → {⟨z, (𝐹g)⟩} V)
72 unexg 4128 . . . 4 ((g V {⟨z, (𝐹g)⟩} V) → (g ∪ {⟨z, (𝐹g)⟩}) V)
7311, 71, 72sylancr 395 . . 3 (φ → (g ∪ {⟨z, (𝐹g)⟩}) V)
744tfrlem3ag 5846 . . 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 616  ∀wal 1226   = wceq 1228   ∈ wcel 1374  {cab 2008   ≠ wne 2186  ∀wral 2284  ∃wrex 2285  Vcvv 2535   ∪ cun 2892   ⊆ wss 2894  {csn 3350  ⟨cop 3353  Ord word 4048  Oncon0 4049  suc csuc 4051  dom cdm 4272   ↾ cres 4274  Fun wfun 4823   Fn wfn 4824  ‘cfv 4829 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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-res 4284  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837 This theorem is referenced by:  tfrlemibacc  5861  tfrlemi14d  5868  tfrlemi14  5869
 Copyright terms: Public domain W3C validator