Theorem foeq2 6685

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

Assertion

Ref	Expression
foeq2	⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶))

Proof of Theorem foeq2

Step	Hyp	Ref	Expression
1		fneq2 6525	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
2	1	anbi1d 630	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶)))
3		df-fo 6439	. 2 ⊢ (𝐹:𝐴–onto→𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶))
4		df-fo 6439	. 2 ⊢ (𝐹:𝐵–onto→𝐶 ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶))
5	2, 3, 4	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ran crn 5590   Fn wfn 6428  –onto→wfo 6431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-cleq 2730  df-fn 6436  df-fo 6439

This theorem is referenced by:  foco  6702  f1oeq2  6705  foeq123d  6709  tposfo  8069  brwdom  9326  brwdom2  9332  canthwdom  9338  cfslb2n  10024  0ramcl  16724  ghmcyg  19497  txcmpb  22795  qtoptopon  22855  fsupprnfi  31026  opidon2OLD  36012  fnfocofob  44571  fullthinc  46327