Theorem f1oeq3 5284

Description: Equality theorem for one-to-one onto functions. (Contributed by set.mm contributors, 10-Feb-1997.)

Assertion

Ref	Expression
f1oeq3	⊢ (A = B → (F:C–1-1-onto→A ↔ F:C–1-1-onto→B))

Proof of Theorem f1oeq3

Step	Hyp	Ref	Expression
1		f1eq3 5256	. . 3 ⊢ (A = B → (F:C–1-1→A ↔ F:C–1-1→B))
2		foeq3 5268	. . 3 ⊢ (A = B → (F:C–onto→A ↔ F:C–onto→B))
3	1, 2	anbi12d 691	. 2 ⊢ (A = B → ((F:C–1-1→A ∧ F:C–onto→A) ↔ (F:C–1-1→B ∧ F:C–onto→B)))
4		df-f1o 4795	. 2 ⊢ (F:C–1-1-onto→A ↔ (F:C–1-1→A ∧ F:C–onto→A))
5		df-f1o 4795	. 2 ⊢ (F:C–1-1-onto→B ↔ (F:C–1-1→B ∧ F:C–onto→B))
6	3, 4, 5	3bitr4g 279	1 ⊢ (A = B → (F:C–1-1-onto→A ↔ F:C–1-1-onto→B))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642  –1-1→wf1 4779  –onto→wfo 4780  –1-1-onto→wf1o 4781

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795

This theorem is used by:  f1oeq23  5285  resdif  5307  resin  5308  f1osng  5324  isoeq5  5487  isoini2  5499  swapres  5513  bren  6031  xpcomen  6053  xpassen  6058  enpw1pw  6076