Theorem f1oeq23 6791

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 6789	. 2 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))
2		f1oeq3 6790	. 2 ⊢ (𝐶 = 𝐷 → (𝐹:𝐵–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷))
3	1, 2	sylan9bb 509	1 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  –1-1-onto→wf1o 6510

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 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2721  df-ss 3931  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518

This theorem is referenced by:  f1ofvswap  7281  enfixsn  9050  ackbij2lem2  10192  seqf1o  14008  eulerthlem2  16752  isgim  19194  islmim  20969  fpwrelmapffs  32657  wrdpmcl  32859  1arithidomlem2  33507  1arithidom  33508  hgt750lemg  34645  poimirlem3  37617  poimirlem15  37629  eldioph2lem1  42748  fundcmpsurbijinj  47411  gricushgr  47917  isgrlim  47981