Theorem f1oeq2 6810

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 6771	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶))
2		foeq2 6790	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶))
3	1, 2	anbi12d 643	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶) ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶)))
4		df-f1o 6544	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐶 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶))
5		df-f1o 6544	. 2 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶))
6	3, 4, 5	3bitr4g 317	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  –1-1→wf1 6534  –onto→wfo 6535  –1-1-onto→wf1o 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544

This theorem is referenced by:  f1oeq23  6812  f1oeq123d  6815  f1oeq2d  6817  resin  6844  isoeq4  7319  breng  8952  f1dmvrnfibi  9298  cnfcom  9669  infxpenc2  10006  fsumf1o  15774  sumsnf  15794  fprodf1o  16000  prodsn  16016  prodsnf  16018  znhash  21677  znunithash  21683  imasf1oxms  24615  wlksnwwlknvbij  30198  clwwlkvbij  30405  eupthp1  30508  derangval  35592  subfacp1lem2a  35605  subfacp1lem3  35607  subfacp1lem5  35609  sumsnd  45672  isuspgrim0lem  48581  isubgr3stgrlem1  48654  usgrexmpl1lem  48709  usgrexmpl2lem  48714  uspgrsprfo  48836  tposf1o  49581