Theorem foeq1 6788

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 6626	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
2		rneq 5926	. . . 4 ⊢ (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
3	2	eqeq1d 2765	. . 3 ⊢ (𝐹 = 𝐺 → (ran 𝐹 = 𝐵 ↔ ran 𝐺 = 𝐵))
4	1, 3	anbi12d 643	. 2 ⊢ (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵)))
5		df-fo 6542	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6		df-fo 6542	. 2 ⊢ (𝐺:𝐴–onto→𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵))
7	4, 5, 6	3bitr4g 317	1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ran crn 5662   Fn wfn 6531  –onto→wfo 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-fun 6538  df-fn 6539  df-fo 6542

This theorem is referenced by:  fimadmfoALT  6803  f1oeq1  6808  foeq123d  6813  resdif  6842  exfo  7100  mapfoss  8845  fodomr  9112  dif1enlem  9140  fodomfir  9283  fowdom  9529  brwdom2  9531  canthp1lem2  10633  mndfo  18811  sursubmefmnd  18950  znzrhfo  21697  pjhfo  32058  elunop  32224  elunop2  32365  symgcom  33403  nnfoctbdjlem  47189  fcoreslem3  47822  fcoresfo  47828  fcoresfob  47829  fundcmpsurbijinjpreimafv  48176