Theorem dff1o4 5450

Description: Alternate definition of one-to-one onto function. (Contributed by NM, 25-Mar-1998.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)

Assertion

Ref	Expression
dff1o4	⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))

Proof of Theorem dff1o4

Step	Hyp	Ref	Expression
1		dff1o2 5447	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
2		3anass 977	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun ◡𝐹 ∧ ran 𝐹 = 𝐵)))
3		df-rn 4622	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
4	3	eqeq1i 2178	. . . . 5 ⊢ (ran 𝐹 = 𝐵 ↔ dom ◡𝐹 = 𝐵)
5	4	anbi2i 454	. . . 4 ⊢ ((Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun ◡𝐹 ∧ dom ◡𝐹 = 𝐵))
6		df-fn 5201	. . . 4 ⊢ (◡𝐹 Fn 𝐵 ↔ (Fun ◡𝐹 ∧ dom ◡𝐹 = 𝐵))
7	5, 6	bitr4i 186	. . 3 ⊢ ((Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ ◡𝐹 Fn 𝐵)
8	7	anbi2i 454	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (Fun ◡𝐹 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
9	1, 2, 8	3bitri 205	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∧ w3a 973   = wceq 1348  ^◡ccnv 4610  dom cdm 4611  ran crn 4612  Fun wfun 5192   Fn wfn 5193  –1-1-onto→wf1o 5197

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-in 3127  df-ss 3134  df-rn 4622  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205

This theorem is referenced by:  f1ocnv  5455  f1oun  5462  f1o00  5477  f1oi  5480  f1osn  5482  f1ompt  5647  f1ofveu  5841  f1ocnvd  6051  f1od2  6214  mapsnf1o2  6674  sbthlemi9  6942  mhmf1o  12693  grpinvf1o  12769  hmeof1o2  13102