Theorem dff1o4 6842

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 6839	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
2		3anass 1096	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun ◡𝐹 ∧ ran 𝐹 = 𝐵)))
3		df-rn 5688	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
4	3	eqeq1i 2738	. . . . 5 ⊢ (ran 𝐹 = 𝐵 ↔ dom ◡𝐹 = 𝐵)
5	4	anbi2i 624	. . . 4 ⊢ ((Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun ◡𝐹 ∧ dom ◡𝐹 = 𝐵))
6		df-fn 6547	. . . 4 ⊢ (◡𝐹 Fn 𝐵 ↔ (Fun ◡𝐹 ∧ dom ◡𝐹 = 𝐵))
7	5, 6	bitr4i 278	. . 3 ⊢ ((Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ ◡𝐹 Fn 𝐵)
8	7	anbi2i 624	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (Fun ◡𝐹 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))
9	1, 2, 8	3bitri 297	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ^◡ccnv 5676  dom cdm 5677  ran crn 5678  Fun wfun 6538   Fn wfn 6539  –1-1-onto→wf1o 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-rn 5688  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551

This theorem is referenced by:  f1ocnv  6846  f1oun  6853  f1o00  6869  f1oi  6872  f1osn  6874  f1oprswap  6878  f1ompt  7111  f1ofveu  7403  f1ocnvd  7657  curry1  8090  curry2  8093  mapsnf1o2  8888  omxpenlem  9073  sbthlem9  9091  compssiso  10369  mptfzshft  15724  invf1o  17716  mhmf1o  18682  grpinvf1o  18893  ghmf1o  19122  rhmf1o  20269  srngf1o  20462  lmhmf1o  20657  hmeof1o2  23267  axcontlem2  28254  f1o3d  31882  padct  31975  f1od2  31977  cdleme51finvN  39475  fsovf1od  42815  isomushgr  46542  mgmhmf1o  46605  rnghmf1o  46749