Theorem f1oeq2 5563

Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997.)

Assertion

Ref	Expression
f1oeq2	⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))

Proof of Theorem f1oeq2

Step	Hyp	Ref	Expression
1		f1eq2 5529	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶))
2		foeq2 5547	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶))
3	1, 2	anbi12d 473	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶) ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶)))
4		df-f1o 5325	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐶 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶))
5		df-f1o 5325	. 2 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶))
6	3, 4, 5	3bitr4g 223	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  –1-1→wf1 5315  –onto→wfo 5316  –1-1-onto→wf1o 5317

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 1493  ax-gen 1495  ax-4 1556  ax-17 1572  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-cleq 2222  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325

This theorem is referenced by:  f1oeq23  5565  f1oeq123d  5568  f1oeq2d  5570  f1osng  5616  isoeq4  5934  breng  6902  bren  6903  f1dmvrnfibi  7122  summodclem3  11906  summodclem2a  11907  summodc  11909  fsum3  11913  fsumf1o  11916  sumsnf  11935  fprodf1o  12114  prodsnf  12118  znfi  14634  znhash  14635