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 1541  –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 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

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 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  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  7249  f1ofvswap  7252  dif1en  9086  cnfcomlem  9608  cnfcom2  9611  cnfcom3clem  9614  infxpenc  9928  infxpenc2lem2  9930  infxpenc2  9932  canthp1lem2  10564  pwfseqlem5  10574  pwfseq  10575  s2f1o  14839  s4f1o  14841  bitsf1ocnv  16371  yonffthlem  18205  grplactcnv  18973  eqgen  19110  znunithash  21519  tgpconncompeqg  24056  fcobijfs  32800  fcobijfs2  32801  indf1o  32946  s2f1  33027  ccatws1f1o  33033  mgcf1o  33085  gsummpt2d  33132  gsumwrd2dccat  33160  subfacp1lem3  35376  subfacp1lem5  35378  ismrer1  38039  hvmap1o  42023  3f1oss2  47322  idfu1stf1o  49344  imaidfu  49355  fucoppc  49655  lmdran  49916