Theorem foeq1 6815

Description: Equality theorem for onto functions. (Contributed by NM, 1-Aug-1994.)

Assertion

Ref	Expression
foeq1	⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))

Proof of Theorem foeq1

Step	Hyp	Ref	Expression
1		fneq1 6658	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
2		rneq 5946	. . . 4 ⊢ (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
3	2	eqeq1d 2738	. . 3 ⊢ (𝐹 = 𝐺 → (ran 𝐹 = 𝐵 ↔ ran 𝐺 = 𝐵))
4	1, 3	anbi12d 632	. 2 ⊢ (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵)))
5		df-fo 6566	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6		df-fo 6566	. 2 ⊢ (𝐺:𝐴–onto→𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵))
7	4, 5, 6	3bitr4g 314	1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ran crn 5685   Fn wfn 6555  –onto→wfo 6558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-fun 6562  df-fn 6563  df-fo 6566

This theorem is referenced by:  fimadmfoALT  6830  f1oeq1  6835  foeq123d  6840  resdif  6868  exfo  7124  mapfoss  8893  fodomr  9169  dif1enlem  9197  dif1enlemOLD  9198  fodomfir  9369  fowdom  9612  brwdom2  9614  canthp1lem2  10694  mndfo  18772  sursubmefmnd  18910  znzrhfo  21567  pjhfo  31726  elunop  31892  elunop2  32033  symgcom  33104  nnfoctbdjlem  46475  fcoreslem3  47082  fcoresfo  47088  fcoresfob  47089  fundcmpsurbijinjpreimafv  47399