dff1o4 - Metamath Proof Explorer

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Theorem dff1o4 6622

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 6619	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵))
2		3anass 1091	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵)))
3		df-rn 5565	. . . . . 6 ⊢ ran 𝐹 = dom ^◡𝐹
4	3	eqeq1i 2826	. . . . 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 6357	. . . 4 ⊢ (^◡𝐹 Fn 𝐵 ↔ (Fun ^◡𝐹 ∧ dom ^◡𝐹 = 𝐵))
7	5, 6	bitr4i 280	. . 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 299	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ^◡𝐹 Fn 𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ^◡ccnv 5553 dom cdm 5554 ran crn 5555 Fun wfun 6348 Fn wfn 6349 –1-1-onto→wf1o 6353

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-in 3942 df-ss 3951 df-rn 5565 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361

This theorem is referenced by: f1ocnv 6626 f1oun 6633 f1o00 6648 f1oi 6651 f1osn 6653 f1oprswap 6657 f1ompt 6874 f1ofveu 7150 f1ocnvd 7395 curry1 7798 curry2 7801 mapsnf1o2 8457 omxpenlem 8617 sbthlem9 8634 compssiso 9795 mptfzshft 15132 fprodrev 15330 invf1o 17038 mhmf1o 17965 grpinvf1o 18168 ghmf1o 18387 rhmf1o 19483 srngf1o 19624 lmhmf1o 19817 hmeof1o2 22370 axcontlem2 26750 f1o3d 30371 padct 30454 f1od2 30456 cdleme51finvN 37691 fsovf1od 40360 isomushgr 43990 mgmhmf1o 44053 rnghmf1o 44173