Theorem foeq3 5436

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

Assertion

Ref	Expression
foeq3	⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵))

Proof of Theorem foeq3

Step	Hyp	Ref	Expression
1		eqeq2 2187	. . 3 ⊢ (𝐴 = 𝐵 → (ran 𝐹 = 𝐴 ↔ ran 𝐹 = 𝐵))
2	1	anbi2d 464	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐴) ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐵)))
3		df-fo 5222	. 2 ⊢ (𝐹:𝐶–onto→𝐴 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐴))
4		df-fo 5222	. 2 ⊢ (𝐹:𝐶–onto→𝐵 ↔ (𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐵))
5	2, 3, 4	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐶–onto→𝐴 ↔ 𝐹:𝐶–onto→𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ran crn 4627   Fn wfn 5211  –onto→wfo 5214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-gen 1449  ax-4 1510  ax-17 1526  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-cleq 2170  df-fo 5222

This theorem is referenced by:  f1oeq3  5451  foeq123d  5454  resdif  5483  ffoss  5493  fifo  6978  enumct  7113  ctssexmid  7147  exmidfodomrlemr  7200  exmidfodomrlemrALT  7201  qnnen  12431  ctiunctal  12441  unct  12442  quslem  12744  subctctexmid  14720