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Theorem tfrlem3ag 5865
Description: Lemma for transfinite recursion. This lemma just changes some bound variables in A for later use. (Contributed by NM, 9-Apr-1995.)
Hypothesis
Ref Expression
tfrlem3.1 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
Assertion
Ref Expression
tfrlem3ag (𝐺 V → (𝐺 Az On (𝐺 Fn z w z (𝐺w) = (𝐹‘(𝐺w)))))
Distinct variable groups:   w,f,x,y,z,𝐹   f,𝐺,w,x,y,z
Allowed substitution hints:   A(x,y,z,w,f)

Proof of Theorem tfrlem3ag
StepHypRef Expression
1 fneq12 4935 . . . 4 ((f = 𝐺 x = z) → (f Fn x𝐺 Fn z))
2 simpll 481 . . . . . . 7 (((f = 𝐺 x = z) y = w) → f = 𝐺)
3 simpr 103 . . . . . . 7 (((f = 𝐺 x = z) y = w) → y = w)
42, 3fveq12d 5127 . . . . . 6 (((f = 𝐺 x = z) y = w) → (fy) = (𝐺w))
52, 3reseq12d 4556 . . . . . . 7 (((f = 𝐺 x = z) y = w) → (fy) = (𝐺w))
65fveq2d 5125 . . . . . 6 (((f = 𝐺 x = z) y = w) → (𝐹‘(fy)) = (𝐹‘(𝐺w)))
74, 6eqeq12d 2051 . . . . 5 (((f = 𝐺 x = z) y = w) → ((fy) = (𝐹‘(fy)) ↔ (𝐺w) = (𝐹‘(𝐺w))))
8 simplr 482 . . . . 5 (((f = 𝐺 x = z) y = w) → x = z)
97, 8cbvraldva2 2531 . . . 4 ((f = 𝐺 x = z) → (y x (fy) = (𝐹‘(fy)) ↔ w z (𝐺w) = (𝐹‘(𝐺w))))
101, 9anbi12d 442 . . 3 ((f = 𝐺 x = z) → ((f Fn x y x (fy) = (𝐹‘(fy))) ↔ (𝐺 Fn z w z (𝐺w) = (𝐹‘(𝐺w)))))
1110cbvrexdva 2534 . 2 (f = 𝐺 → (x On (f Fn x y x (fy) = (𝐹‘(fy))) ↔ z On (𝐺 Fn z w z (𝐺w) = (𝐹‘(𝐺w)))))
12 tfrlem3.1 . 2 A = {fx On (f Fn x y x (fy) = (𝐹‘(fy)))}
1311, 12elab2g 2683 1 (𝐺 V → (𝐺 Az On (𝐺 Fn z w z (𝐺w) = (𝐹‘(𝐺w)))))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  Vcvv 2551  Oncon0 4066  cres 4290   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-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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  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:  tfrlemisucaccv  5880
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