Theorem dff1o4 6788

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 6785	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵))
2		3anass 1095	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun ◡𝐹 ∧ ran 𝐹 = 𝐵)))
3		df-rn 5642	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
4	3	eqeq1i 2741	. . . . 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 6501	. . . 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 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ^◡ccnv 5630  dom cdm 5631  ran crn 5632  Fun wfun 6492   Fn wfn 6493  –1-1-onto→wf1o 6497

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-ex 1782  df-cleq 2728  df-ss 3906  df-rn 5642  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505

This theorem is referenced by:  f1ocnv  6792  f1oun  6799  f1o00  6815  f1oiOLD  6819  f1osn  6821  f1oprswap  6825  f1ompt  7063  f1ofveu  7361  f1ocnvd  7618  curry1  8054  curry2  8057  mapsnf1o2  8842  omxpenlem  9016  sbthlem9  9033  compssiso  10296  mptfzshft  15740  invf1o  17736  mgmhmf1o  18668  mhmf1o  18764  grpinvf1o  18985  ghmf1o  19223  rnghmf1o  20432  rhmf1o  20470  srngf1o  20825  lmhmf1o  21041  hmeof1o2  23728  axcontlem2  29034  f1o3d  32699  padct  32791  f1od2  32792  cdleme51finvN  41002  fsovf1od  44443  gricushgr  48393  imaf1homlem  49582  idemb  49634