Theorem f1oeq23 6809

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 6807	. 2 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))
2		f1oeq3 6808	. 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 6530

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 2007  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2727  df-ss 3943  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538

This theorem is referenced by:  f1ofvswap  7299  enfixsn  9095  ackbij2lem2  10253  seqf1o  14061  eulerthlem2  16801  isgim  19245  islmim  21020  fpwrelmapffs  32711  wrdpmcl  32913  1arithidomlem2  33551  1arithidom  33552  hgt750lemg  34686  poimirlem3  37647  poimirlem15  37659  eldioph2lem1  42783  fundcmpsurbijinj  47424  gricushgr  47930  isgrlim  47994