Theorem f1oeq2 6764

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 6727	. . 3 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1→𝐶 ↔ 𝐹:𝐵–1-1→𝐶))
2		foeq2 6744	. . 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 6500	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐶 ↔ (𝐹:𝐴–1-1→𝐶 ∧ 𝐹:𝐴–onto→𝐶))
5		df-f1o 6500	. 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 6490  –onto→wfo 6491  –1-1-onto→wf1o 6492

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 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500

This theorem is referenced by:  f1oeq23  6766  f1oeq123d  6769  f1oeq2d  6771  resin  6797  isoeq4  7269  breng  8896  f1dmvrnfibi  9245  cnfcom  9615  infxpenc2  9938  fsumf1o  15679  sumsnf  15699  fprodf1o  15905  prodsn  15921  prodsnf  15923  znhash  21551  znunithash  21557  imasf1oxms  24467  wlksnwwlknvbij  29994  clwwlkvbij  30201  eupthp1  30304  derangval  35368  subfacp1lem2a  35381  subfacp1lem3  35383  subfacp1lem5  35385  sumsnd  45478  isuspgrim0lem  48384  isubgr3stgrlem1  48457  usgrexmpl1lem  48512  usgrexmpl2lem  48517  uspgrsprfo  48639  tposf1o  49374