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Theorem tfrlemi1 5887
Description: We can define an acceptable function on any ordinal.

As with many of the transfinite recursion theorems, we have a hypothesis that states that 𝐹 is a function and that it is defined for all ordinals. (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))
Assertion
Ref Expression
tfrlemi1 ((φ 𝐶 On) → g(g Fn 𝐶 u 𝐶 (gu) = (𝐹‘(gu))))
Distinct variable groups:   f,g,u,x,y,A   f,𝐹,g,u,x,y   φ,y   𝐶,g,u   φ,f
Allowed substitution hints:   φ(x,u,g)   𝐶(x,y,f)

Proof of Theorem tfrlemi1
Dummy variables 𝑒 𝑘 𝑡 v w z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 103 . . . . . . 7 ((z = w g = 𝑘) → g = 𝑘)
2 simpl 102 . . . . . . 7 ((z = w g = 𝑘) → z = w)
31, 2fneq12d 4934 . . . . . 6 ((z = w g = 𝑘) → (g Fn z𝑘 Fn w))
41fveq1d 5123 . . . . . . . 8 ((z = w g = 𝑘) → (gu) = (𝑘u))
51reseq1d 4554 . . . . . . . . 9 ((z = w g = 𝑘) → (gu) = (𝑘u))
65fveq2d 5125 . . . . . . . 8 ((z = w g = 𝑘) → (𝐹‘(gu)) = (𝐹‘(𝑘u)))
74, 6eqeq12d 2051 . . . . . . 7 ((z = w g = 𝑘) → ((gu) = (𝐹‘(gu)) ↔ (𝑘u) = (𝐹‘(𝑘u))))
82, 7raleqbidv 2511 . . . . . 6 ((z = w g = 𝑘) → (u z (gu) = (𝐹‘(gu)) ↔ u w (𝑘u) = (𝐹‘(𝑘u))))
93, 8anbi12d 442 . . . . 5 ((z = w g = 𝑘) → ((g Fn z u z (gu) = (𝐹‘(gu))) ↔ (𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u)))))
109cbvexdva 1801 . . . 4 (z = w → (g(g Fn z u z (gu) = (𝐹‘(gu))) ↔ 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u)))))
1110imbi2d 219 . . 3 (z = w → ((φg(g Fn z u z (gu) = (𝐹‘(gu)))) ↔ (φ𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u))))))
12 fneq2 4931 . . . . . 6 (z = 𝐶 → (g Fn zg Fn 𝐶))
13 raleq 2499 . . . . . 6 (z = 𝐶 → (u z (gu) = (𝐹‘(gu)) ↔ u 𝐶 (gu) = (𝐹‘(gu))))
1412, 13anbi12d 442 . . . . 5 (z = 𝐶 → ((g Fn z u z (gu) = (𝐹‘(gu))) ↔ (g Fn 𝐶 u 𝐶 (gu) = (𝐹‘(gu)))))
1514exbidv 1703 . . . 4 (z = 𝐶 → (g(g Fn z u z (gu) = (𝐹‘(gu))) ↔ g(g Fn 𝐶 u 𝐶 (gu) = (𝐹‘(gu)))))
1615imbi2d 219 . . 3 (z = 𝐶 → ((φg(g Fn z u z (gu) = (𝐹‘(gu)))) ↔ (φg(g Fn 𝐶 u 𝐶 (gu) = (𝐹‘(gu))))))
17 r19.21v 2390 . . . 4 (w z (φ𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u)))) ↔ (φw z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u)))))
18 tfrlemisucfn.1 . . . . . . . . 9 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
1918tfrlem3 5867 . . . . . . . 8 A = {gz On (g Fn z 𝑒 z (g𝑒) = (𝐹‘(g𝑒)))}
20 tfrlemisucfn.2 . . . . . . . . . 10 (φx(Fun 𝐹 (𝐹x) V))
21 fveq2 5121 . . . . . . . . . . . . 13 (x = z → (𝐹x) = (𝐹z))
2221eleq1d 2103 . . . . . . . . . . . 12 (x = z → ((𝐹x) V ↔ (𝐹z) V))
2322anbi2d 437 . . . . . . . . . . 11 (x = z → ((Fun 𝐹 (𝐹x) V) ↔ (Fun 𝐹 (𝐹z) V)))
2423cbvalv 1791 . . . . . . . . . 10 (x(Fun 𝐹 (𝐹x) V) ↔ z(Fun 𝐹 (𝐹z) V))
2520, 24sylib 127 . . . . . . . . 9 (φz(Fun 𝐹 (𝐹z) V))
2625adantr 261 . . . . . . . 8 ((φ (z On w z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u))))) → z(Fun 𝐹 (𝐹z) V))
27 simpr 103 . . . . . . . . . . . . 13 (((𝑡 = w = v) 𝑘 = f) → 𝑘 = f)
28 simplr 482 . . . . . . . . . . . . 13 (((𝑡 = w = v) 𝑘 = f) → w = v)
2927, 28fneq12d 4934 . . . . . . . . . . . 12 (((𝑡 = w = v) 𝑘 = f) → (𝑘 Fn wf Fn v))
3027eleq1d 2103 . . . . . . . . . . . 12 (((𝑡 = w = v) 𝑘 = f) → (𝑘 Af A))
31 simpll 481 . . . . . . . . . . . . 13 (((𝑡 = w = v) 𝑘 = f) → 𝑡 = )
3227fveq2d 5125 . . . . . . . . . . . . . . . 16 (((𝑡 = w = v) 𝑘 = f) → (𝐹𝑘) = (𝐹f))
3328, 32opeq12d 3548 . . . . . . . . . . . . . . 15 (((𝑡 = w = v) 𝑘 = f) → ⟨w, (𝐹𝑘)⟩ = ⟨v, (𝐹f)⟩)
3433sneqd 3380 . . . . . . . . . . . . . 14 (((𝑡 = w = v) 𝑘 = f) → {⟨w, (𝐹𝑘)⟩} = {⟨v, (𝐹f)⟩})
3527, 34uneq12d 3092 . . . . . . . . . . . . 13 (((𝑡 = w = v) 𝑘 = f) → (𝑘 ∪ {⟨w, (𝐹𝑘)⟩}) = (f ∪ {⟨v, (𝐹f)⟩}))
3631, 35eqeq12d 2051 . . . . . . . . . . . 12 (((𝑡 = w = v) 𝑘 = f) → (𝑡 = (𝑘 ∪ {⟨w, (𝐹𝑘)⟩}) ↔ = (f ∪ {⟨v, (𝐹f)⟩})))
3729, 30, 363anbi123d 1206 . . . . . . . . . . 11 (((𝑡 = w = v) 𝑘 = f) → ((𝑘 Fn w 𝑘 A 𝑡 = (𝑘 ∪ {⟨w, (𝐹𝑘)⟩})) ↔ (f Fn v f A = (f ∪ {⟨v, (𝐹f)⟩}))))
3837cbvexdva 1801 . . . . . . . . . 10 ((𝑡 = w = v) → (𝑘(𝑘 Fn w 𝑘 A 𝑡 = (𝑘 ∪ {⟨w, (𝐹𝑘)⟩})) ↔ f(f Fn v f A = (f ∪ {⟨v, (𝐹f)⟩}))))
3938cbvrexdva 2534 . . . . . . . . 9 (𝑡 = → (w z 𝑘(𝑘 Fn w 𝑘 A 𝑡 = (𝑘 ∪ {⟨w, (𝐹𝑘)⟩})) ↔ v z f(f Fn v f A = (f ∪ {⟨v, (𝐹f)⟩}))))
4039cbvabv 2158 . . . . . . . 8 {𝑡w z 𝑘(𝑘 Fn w 𝑘 A 𝑡 = (𝑘 ∪ {⟨w, (𝐹𝑘)⟩}))} = {v z f(f Fn v f A = (f ∪ {⟨v, (𝐹f)⟩}))}
41 simpl 102 . . . . . . . . 9 ((z On w z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u)))) → z On)
4241adantl 262 . . . . . . . 8 ((φ (z On w z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u))))) → z On)
43 simpr 103 . . . . . . . . . 10 ((z On w z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u)))) → w z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u))))
44 simpr 103 . . . . . . . . . . . . . 14 ((w = v 𝑘 = f) → 𝑘 = f)
45 simpl 102 . . . . . . . . . . . . . 14 ((w = v 𝑘 = f) → w = v)
4644, 45fneq12d 4934 . . . . . . . . . . . . 13 ((w = v 𝑘 = f) → (𝑘 Fn wf Fn v))
47 simplr 482 . . . . . . . . . . . . . . . 16 (((w = v 𝑘 = f) u = y) → 𝑘 = f)
48 simpr 103 . . . . . . . . . . . . . . . 16 (((w = v 𝑘 = f) u = y) → u = y)
4947, 48fveq12d 5127 . . . . . . . . . . . . . . 15 (((w = v 𝑘 = f) u = y) → (𝑘u) = (fy))
5047, 48reseq12d 4556 . . . . . . . . . . . . . . . 16 (((w = v 𝑘 = f) u = y) → (𝑘u) = (fy))
5150fveq2d 5125 . . . . . . . . . . . . . . 15 (((w = v 𝑘 = f) u = y) → (𝐹‘(𝑘u)) = (𝐹‘(fy)))
5249, 51eqeq12d 2051 . . . . . . . . . . . . . 14 (((w = v 𝑘 = f) u = y) → ((𝑘u) = (𝐹‘(𝑘u)) ↔ (fy) = (𝐹‘(fy))))
53 simpll 481 . . . . . . . . . . . . . 14 (((w = v 𝑘 = f) u = y) → w = v)
5452, 53cbvraldva2 2531 . . . . . . . . . . . . 13 ((w = v 𝑘 = f) → (u w (𝑘u) = (𝐹‘(𝑘u)) ↔ y v (fy) = (𝐹‘(fy))))
5546, 54anbi12d 442 . . . . . . . . . . . 12 ((w = v 𝑘 = f) → ((𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u))) ↔ (f Fn v y v (fy) = (𝐹‘(fy)))))
5655cbvexdva 1801 . . . . . . . . . . 11 (w = v → (𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u))) ↔ f(f Fn v y v (fy) = (𝐹‘(fy)))))
5756cbvralv 2527 . . . . . . . . . 10 (w z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u))) ↔ v z f(f Fn v y v (fy) = (𝐹‘(fy))))
5843, 57sylib 127 . . . . . . . . 9 ((z On w z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u)))) → v z f(f Fn v y v (fy) = (𝐹‘(fy))))
5958adantl 262 . . . . . . . 8 ((φ (z On w z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u))))) → v z f(f Fn v y v (fy) = (𝐹‘(fy))))
6019, 26, 40, 42, 59tfrlemiex 5886 . . . . . . 7 ((φ (z On w z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u))))) → g(g Fn z u z (gu) = (𝐹‘(gu))))
6160expr 357 . . . . . 6 ((φ z On) → (w z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u))) → g(g Fn z u z (gu) = (𝐹‘(gu)))))
6261expcom 109 . . . . 5 (z On → (φ → (w z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u))) → g(g Fn z u z (gu) = (𝐹‘(gu))))))
6362a2d 23 . . . 4 (z On → ((φw z 𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u)))) → (φg(g Fn z u z (gu) = (𝐹‘(gu))))))
6417, 63syl5bi 141 . . 3 (z On → (w z (φ𝑘(𝑘 Fn w u w (𝑘u) = (𝐹‘(𝑘u)))) → (φg(g Fn z u z (gu) = (𝐹‘(gu))))))
6511, 16, 64tfis3 4252 . 2 (𝐶 On → (φg(g Fn 𝐶 u 𝐶 (gu) = (𝐹‘(gu)))))
6665impcom 116 1 ((φ 𝐶 On) → g(g Fn 𝐶 u 𝐶 (gu) = (𝐹‘(gu))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884  wal 1240   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wral 2300  wrex 2301  Vcvv 2551  cun 2909  {csn 3367  cop 3370  Oncon0 4066  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-coll 3863  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-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  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-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  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-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861
This theorem is referenced by:  tfrlemi14d  5888  tfrexlem  5889
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