Theorem foeq1 5122

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 5007	. . 3 ⊢ (𝐹 = 𝐺 → (𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴))
2		rneq 4579	. . . 4 ⊢ (𝐹 = 𝐺 → ran 𝐹 = ran 𝐺)
3	2	eqeq1d 2089	. . 3 ⊢ (𝐹 = 𝐺 → (ran 𝐹 = 𝐵 ↔ ran 𝐺 = 𝐵))
4	1, 3	anbi12d 456	. 2 ⊢ (𝐹 = 𝐺 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵) ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵)))
5		df-fo 4928	. 2 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐵))
6		df-fo 4928	. 2 ⊢ (𝐺:𝐴–onto→𝐵 ↔ (𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝐵))
7	4, 5, 6	3bitr4g 221	1 ⊢ (𝐹 = 𝐺 → (𝐹:𝐴–onto→𝐵 ↔ 𝐺:𝐴–onto→𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  ran crn 4364   Fn wfn 4917  –onto→wfo 4920

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-fun 4924  df-fn 4925  df-fo 4928

This theorem is referenced by:  f1oeq1  5137  foeq123d  5142  resdif  5168  dif1en  6364