Theorem f1oeq23 5285

Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012.)

Assertion

Ref	Expression
f1oeq23	⊢ ((A = B ∧ C = D) → (F:A–1-1-onto→C ↔ F:B–1-1-onto→D))

Proof of Theorem f1oeq23

Step	Hyp	Ref	Expression
1		f1oeq2 5283	. 2 ⊢ (A = B → (F:A–1-1-onto→C ↔ F:B–1-1-onto→C))
2		f1oeq3 5284	. 2 ⊢ (C = D → (F:B–1-1-onto→C ↔ F:B–1-1-onto→D))
3	1, 2	sylan9bb 680	1 ⊢ ((A = B ∧ C = D) → (F:A–1-1-onto→C ↔ F:B–1-1-onto→D))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  –1-1-onto→wf1o 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795

This theorem is referenced by: (None)