Theorem f1oeq1d 6857

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 6850	. 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 1537  –1-1-onto→wf1o 6572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580

This theorem is referenced by:  f1orescnv  6877  f1osng  6903  f1ocoima  7339  f1ofvswap  7342  dif1en  9226  dif1enOLD  9228  cnfcomlem  9768  cnfcom2  9771  cnfcom3clem  9774  infxpenc  10087  infxpenc2lem2  10089  infxpenc2  10091  canthp1lem2  10722  pwfseqlem5  10732  pwfseq  10733  s2f1o  14965  s4f1o  14967  bitsf1ocnv  16490  yonffthlem  18352  grplactcnv  19083  eqgen  19221  znunithash  21606  tgpconncompeqg  24141  fcobijfs  32737  s2f1  32911  ccatws1f1o  32918  mgcf1o  32976  gsummpt2d  33032  indf1o  33988  subfacp1lem3  35150  subfacp1lem5  35152  ismrer1  37798  hvmap1o  41720  metakunt34  42195  3f1oss2  46991