MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  recseq Structured version   Visualization version   GIF version

Theorem recseq 7430
Description: Equality theorem for recs. (Contributed by Stefan O'Rear, 18-Jan-2015.)
Assertion
Ref Expression
recseq (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))

Proof of Theorem recseq
StepHypRef Expression
1 wrecseq3 7372 . 2 (𝐹 = 𝐺 → wrecs( E , On, 𝐹) = wrecs( E , On, 𝐺))
2 df-recs 7428 . 2 recs(𝐹) = wrecs( E , On, 𝐹)
3 df-recs 7428 . 2 recs(𝐺) = wrecs( E , On, 𝐺)
41, 2, 33eqtr4g 2680 1 (𝐹 = 𝐺 → recs(𝐹) = recs(𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480   E cep 4993  Oncon0 5692  wrecscwrecs 7366  recscrecs 7427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-iota 5820  df-fv 5865  df-wrecs 7367  df-recs 7428
This theorem is referenced by:  rdgeq1  7467  rdgeq2  7468  dfoi  8376  oieq1  8377  oieq2  8378  ordtypecbv  8382  dfac12r  8928  zorn2g  9285  ttukey2g  9298  csbrdgg  32846  aomclem3  37145  aomclem8  37150
  Copyright terms: Public domain W3C validator