Theorem f1oeq1d 6769

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 6762	. 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 1542  –1-1-onto→wf1o 6491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499

This theorem is referenced by:  f1orescnv  6789  f1osng  6816  f1ocoima  7251  f1ofvswap  7254  dif1en  9089  cnfcomlem  9611  cnfcom2  9614  cnfcom3clem  9617  infxpenc  9931  infxpenc2lem2  9933  infxpenc2  9935  canthp1lem2  10567  pwfseqlem5  10577  pwfseq  10578  s2f1o  14869  s4f1o  14871  bitsf1ocnv  16404  yonffthlem  18239  grplactcnv  19010  eqgen  19147  znunithash  21554  tgpconncompeqg  24087  fcobijfs  32809  fcobijfs2  32810  indf1o  32939  s2f1  33020  ccatws1f1o  33026  mgcf1o  33078  gsummpt2d  33125  gsumwrd2dccat  33154  subfacp1lem3  35380  subfacp1lem5  35382  ismrer1  38173  hvmap1o  42223  3f1oss2  47536  idfu1stf1o  49586  imaidfu  49597  fucoppc  49897  lmdran  50158