Theorem foeq1 6822

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

Assertion

Ref	Expression
foeq1	⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶onto𝐵 ↔ 𝐺:𝐴⟶onto𝐵))

Proof of Theorem foeq1

Step	Hyp	Ref	Expression
1		fneq1 6660	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
2		rneq 5930	. . . 4 ⊢ (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
3	2	eqeq1d 2765	. . 3 ⊢ (𝐹 = 𝐺 → (ran 𝐹 = 𝐵 ↔ ran 𝐺 = 𝐵))
4	1, 3	anbi12d 644	. 2 ⊢ (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵)))
5		df-fo 6572	. 2 ⊢ (𝐹:𝐴⟶onto𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6		df-fo 6572	. 2 ⊢ (𝐺:𝐴⟶onto𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵))
7	4, 5, 6	3bitr4g 317	1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴⟶onto𝐵 ↔ 𝐺:𝐴⟶onto𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ran crn 5659   Fn wfn 6561  ⟶ontowfo 6564

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-fo 6572

This theorem is used by:  fimadmfoALT  6837  f1oeq1  6842  foeq123d  6847  resdif  6876  exfo  7137  mapfoss  8912  fodomr  9185  dif1enlem  9213  fodomfir  9357  fowdom  9602  brwdom2  9604  canthp1lem2  10777  mgmfod  18909  mndfoOLD  19000  sursubmefmnd  19142  znzrhfo  21908  pjhfo  32365  elunop  32531  elunop2  32672  symgcom  33743  nnfoctbdjlem  47738  fcoreslem3  48408  fcoresfo  48414  fcoresfob  48415  fundcmpsurbijinjpreimafv  48762