Theorem f1oeq1d 6695

Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
f1oeq1d.1	⊢ (𝜑 → 𝐹 = 𝐺)

Assertion

Ref	Expression
f1oeq1d	⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵))

Proof of Theorem f1oeq1d

Step	Hyp	Ref	Expression
1		f1oeq1d.1	. 2 ⊢ (𝜑 → 𝐹 = 𝐺)
2		f1oeq1 6688	. 2 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539  –1-1-onto→wf1o 6417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425

This theorem is referenced by:  f1orescnv  6715  f1osng  6740  f1ofvswap  7158  dif1en  8907  cnfcomlem  9387  cnfcom2  9390  cnfcom3clem  9393  infxpenc  9705  infxpenc2lem2  9707  infxpenc2  9709  canthp1lem2  10340  pwfseqlem5  10350  pwfseq  10351  s2f1o  14557  s4f1o  14559  bitsf1ocnv  16079  yonffthlem  17916  grplactcnv  18593  eqgen  18724  znunithash  20684  tgpconncompeqg  23171  fcobijfs  30960  s2f1  31121  mgcf1o  31183  gsummpt2d  31211  indf1o  31892  subfacp1lem3  33044  subfacp1lem5  33046  ismrer1  35923  hvmap1o  39704  metakunt34  40086