Theorem f1oeq1d 6777

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 6770	. 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 1540  –1-1-onto→wf1o 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506

This theorem is referenced by:  f1orescnv  6797  f1osng  6823  f1ocoima  7260  f1ofvswap  7263  dif1en  9101  dif1enOLD  9103  cnfcomlem  9628  cnfcom2  9631  cnfcom3clem  9634  infxpenc  9947  infxpenc2lem2  9949  infxpenc2  9951  canthp1lem2  10582  pwfseqlem5  10592  pwfseq  10593  s2f1o  14858  s4f1o  14860  bitsf1ocnv  16390  yonffthlem  18219  grplactcnv  18951  eqgen  19089  znunithash  21450  tgpconncompeqg  23975  fcobijfs  32619  indf1o  32760  s2f1  32839  ccatws1f1o  32846  mgcf1o  32902  gsummpt2d  32962  gsumwrd2dccat  32980  subfacp1lem3  35142  subfacp1lem5  35144  ismrer1  37805  hvmap1o  41730  3f1oss2  47050  idfu1stf1o  49061  imaidfu  49072  fucoppc  49372  lmdran  49633