Theorem f1oeq1d 6753

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 6746	. 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 206   = wceq 1541  –1-1-onto→wf1o 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4279  df-if 4471  df-sn 4572  df-pr 4574  df-op 4578  df-br 5087  df-opab 5149  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483

This theorem is referenced by:  f1orescnv  6773  f1osng  6799  f1ocoima  7232  f1ofvswap  7235  dif1en  9066  cnfcomlem  9584  cnfcom2  9587  cnfcom3clem  9590  infxpenc  9904  infxpenc2lem2  9906  infxpenc2  9908  canthp1lem2  10539  pwfseqlem5  10549  pwfseq  10550  s2f1o  14818  s4f1o  14820  bitsf1ocnv  16350  yonffthlem  18183  grplactcnv  18951  eqgen  19088  znunithash  21496  tgpconncompeqg  24022  fcobijfs  32696  fcobijfs2  32697  indf1o  32837  s2f1  32918  ccatws1f1o  32924  mgcf1o  32976  gsummpt2d  33021  gsumwrd2dccat  33039  subfacp1lem3  35218  subfacp1lem5  35220  ismrer1  37878  hvmap1o  41802  3f1oss2  47107  idfu1stf1o  49131  imaidfu  49142  fucoppc  49442  lmdran  49703