Theorem f1oeq1d 6796

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 6789	. 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 208   = wceq 1559  –1-1-onto→wf1o 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523

This theorem is referenced by:  f1orescnv  6817  f1osng  6844  f1ocoima  7282  f1ofvswap  7285  dif1en  9124  cnfcomlem  9648  cnfcom2  9651  cnfcom3clem  9654  infxpenc  9968  infxpenc2lem2  9970  infxpenc2  9972  canthp1lem2  10605  pwfseqlem5  10615  pwfseq  10616  s2f1o  14923  s4f1o  14925  bitsf1ocnv  16469  yonffthlem  18305  grplactcnv  19076  eqgen  19213  znunithash  21604  tgpconncompeqg  24160  fcobijfs  32884  fcobijfs2  32885  indf1o  33003  s2f1  33084  ccatws1f1o  33090  mgcf1o  33142  gsummpt2d  33190  gsumwrd2dccat  33219  subfacp1lem3  35493  subfacp1lem5  35495  ismrer1  38298  hvmap1o  42348  3f1oss2  47631  idfu1stf1o  49681  imaidfu  49692  fucoppc  49992  lmdran  50253