Theorem f1oeq1d 6849

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⟶1-1onto𝐵 ↔ 𝐺:𝐴⟶1-1onto𝐵))

Proof of Theorem f1oeq1d

Step	Hyp	Ref	Expression
1		f1oeq1d.1	. 2 ⊢ (𝜑 → 𝐹 = 𝐺)
2		f1oeq1 6842	. 2 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶1-1onto𝐵 ↔ 𝐺:𝐴⟶1-1onto𝐵))
3	1, 2	syl 18	1 ⊢ (𝜑 → (𝐹:𝐴⟶1-1onto𝐵 ↔ 𝐺:𝐴⟶1-1onto𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570  ⟶1-1ontowf1o 6565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573

This theorem is used by:  f1orescnv  6870  f1osng  6898  f1ocoima  7342  f1ofvswap  7345  dif1en  9215  cnfcomlem  9737  cnfcom2  9740  cnfcom3clem  9743  infxpenc  10136  infxpenc2lem2  10138  infxpenc2  10140  canthp1lem2  10777  pwfseqlem5  10787  pwfseq  10788  s2f1o  15110  s4f1o  15112  bitsf1ocnv  16658  yonffthlem  18506  grplactcnv  19303  eqgen  19443  znunithash  21925  tgpconncompeqg  24486  fcobijfs  33409  fcobijfs2  33410  indf1o  33527  s2f1  33606  ccatws1f1o  33610  mgcf1o  33663  gsummpt2d  33709  gsumwrd2dccat  33738  vonf1onprcf1ac  36085  subfacp1lem3  36155  subfacp1lem5  36157  ismrer1  39038  hvmap1o  43086  3f1oss2  48419  idfu1stf1o  50546  imaidfu  50557  fucoppc  50876  lmdran  51185