Theorem f1oeq1d 6815

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 6808	. 2 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵))
3	1, 2	syl 18	1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ↔ 𝐺:𝐴–1-1-onto→𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570  –1-1-onto→wf1o 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543

This theorem is referenced by:  f1orescnv  6836  f1osng  6863  f1ocoima  7301  f1ofvswap  7304  dif1en  9142  cnfcomlem  9664  cnfcom2  9667  cnfcom3clem  9670  infxpenc  9998  infxpenc2lem2  10000  infxpenc2  10002  canthp1lem2  10633  pwfseqlem5  10643  pwfseq  10644  s2f1o  14949  s4f1o  14951  bitsf1ocnv  16497  yonffthlem  18333  grplactcnv  19104  eqgen  19244  znunithash  21714  tgpconncompeqg  24269  fcobijfs  33066  fcobijfs2  33067  indf1o  33184  s2f1  33265  ccatws1f1o  33271  mgcf1o  33323  gsummpt2d  33369  gsumwrd2dccat  33398  subfacp1lem3  35674  subfacp1lem5  35676  ismrer1  38489  hvmap1o  42537  3f1oss2  47813  idfu1stf1o  49877  imaidfu  49888  fucoppc  50188  lmdran  50449