Theorem f1oeq1d 6766

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 6759	. 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 6488

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 2705

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 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496

This theorem is referenced by:  f1orescnv  6786  f1osng  6813  f1ocoima  7246  f1ofvswap  7249  dif1en  9082  cnfcomlem  9600  cnfcom2  9603  cnfcom3clem  9606  infxpenc  9920  infxpenc2lem2  9922  infxpenc2  9924  canthp1lem2  10555  pwfseqlem5  10565  pwfseq  10566  s2f1o  14830  s4f1o  14832  bitsf1ocnv  16362  yonffthlem  18196  grplactcnv  18964  eqgen  19101  znunithash  21510  tgpconncompeqg  24047  fcobijfs  32728  fcobijfs2  32729  indf1o  32874  s2f1  32955  ccatws1f1o  32961  mgcf1o  33013  gsummpt2d  33060  gsumwrd2dccat  33088  subfacp1lem3  35298  subfacp1lem5  35300  ismrer1  37951  hvmap1o  41935  3f1oss2  47238  idfu1stf1o  49260  imaidfu  49271  fucoppc  49571  lmdran  49832