Theorem foeq1 6732

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 6573	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
2		rneq 5878	. . . 4 ⊢ (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
3	2	eqeq1d 2731	. . 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 6488	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6		df-fo 6488	. 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 1540  ran crn 5620   Fn wfn 6477  –onto→wfo 6480

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 3395  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-fun 6484  df-fn 6485  df-fo 6488

This theorem is referenced by:  fimadmfoALT  6747  f1oeq1  6752  foeq123d  6757  resdif  6785  exfo  7039  mapfoss  8779  fodomr  9045  dif1enlem  9073  fodomfir  9218  fowdom  9463  brwdom2  9465  canthp1lem2  10547  mndfo  18632  sursubmefmnd  18770  znzrhfo  21454  pjhfo  31650  elunop  31816  elunop2  31957  symgcom  33025  nnfoctbdjlem  46440  fcoreslem3  47053  fcoresfo  47059  fcoresfob  47060  fundcmpsurbijinjpreimafv  47395