Theorem f1oeq23 6825

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 6823	. 2 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))
2		f1oeq3 6824	. 2 ⊢ (𝐶 = 𝐷 → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷))
3	1, 2	sylan9bb 511	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  –1-1-onto→wf1o 6543

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551

This theorem is referenced by:  f1ofvswap  7304  enfixsn  9081  ackbij2lem2  10235  seqf1o  14009  eulerthlem2  16715  isgim  19136  islmim  20673  fpwrelmapffs  31990  hgt750lemg  33697  poimirlem3  36539  poimirlem15  36551  eldioph2lem1  41546  fundcmpsurbijinj  46126  isomushgr  46542