Theorem f1oeq1d 6775

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 6768	. 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 6497

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 2708

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 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505

This theorem is referenced by:  f1orescnv  6795  f1osng  6822  f1ocoima  7258  f1ofvswap  7261  dif1en  9096  cnfcomlem  9620  cnfcom2  9623  cnfcom3clem  9626  infxpenc  9940  infxpenc2lem2  9942  infxpenc2  9944  canthp1lem2  10576  pwfseqlem5  10586  pwfseq  10587  s2f1o  14878  s4f1o  14880  bitsf1ocnv  16413  yonffthlem  18248  grplactcnv  19019  eqgen  19156  znunithash  21544  tgpconncompeqg  24077  fcobijfs  32794  fcobijfs2  32795  indf1o  32924  s2f1  33005  ccatws1f1o  33011  mgcf1o  33063  gsummpt2d  33110  gsumwrd2dccat  33139  subfacp1lem3  35364  subfacp1lem5  35366  ismrer1  38159  hvmap1o  42209  3f1oss2  47524  idfu1stf1o  49574  imaidfu  49585  fucoppc  49885  lmdran  50146