Theorem f1oeq2 6771

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 6734	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶))
2		foeq2 6751	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–onto→𝐶 ↔ 𝐹:𝐵–onto→𝐶))
3	1, 2	anbi12d 633	. 2 ⊢ (𝐴 = 𝐵 → ((𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶) ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶)))
4		df-f1o 6507	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐶 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶))
5		df-f1o 6507	. 2 ⊢ (𝐹:𝐵–1-1-onto→𝐶 ↔ (𝐹:𝐵–1-1→𝐶 ∧ 𝐹:𝐵–onto→𝐶))
6	3, 4, 5	3bitr4g 314	1 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  –1-1→wf1 6497  –onto→wfo 6498  –1-1-onto→wf1o 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507

This theorem is referenced by:  f1oeq23  6773  f1oeq123d  6776  f1oeq2d  6778  resin  6804  isoeq4  7276  breng  8904  f1dmvrnfibi  9253  cnfcom  9621  infxpenc2  9944  fsumf1o  15658  sumsnf  15678  fprodf1o  15881  prodsn  15897  prodsnf  15899  znhash  21528  znunithash  21534  imasf1oxms  24448  wlksnwwlknvbij  29997  clwwlkvbij  30204  eupthp1  30307  derangval  35387  subfacp1lem2a  35400  subfacp1lem3  35402  subfacp1lem5  35404  sumsnd  45390  isuspgrim0lem  48257  isubgr3stgrlem1  48330  usgrexmpl1lem  48385  usgrexmpl2lem  48390  uspgrsprfo  48512  tposf1o  49247