Theorem foeq2 6787

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 6630	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵))
2	1	anbi1d 631	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶) ↔ (𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶)))
3		df-fo 6537	. 2 ⊢ (𝐹:𝐴–onto→𝐶 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶))
4		df-fo 6537	. 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 206   ∧ wa 395   = wceq 1540  ran crn 5655   Fn wfn 6526  –onto→wfo 6529

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 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2727  df-fn 6534  df-fo 6537

This theorem is referenced by:  foco  6804  f1oeq2  6807  foeq123d  6811  tposfo  8252  brwdom  9581  brwdom2  9587  canthwdom  9593  cfslb2n  10282  0ramcl  17043  ghmcyg  19877  txcmpb  23582  qtoptopon  23642  fsupprnfi  32669  opidon2OLD  37878  fnfocofob  47108  fullthinc  49336