dff1o4 - Metamath Proof Explorer

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Theorem dff1o4 6306

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 6303	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵))
2		3anass 1081	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵)))
3		df-rn 5277	. . . . . 6 ⊢ ran 𝐹 = dom ^◡𝐹
4	3	eqeq1i 2765	. . . . 5 ⊢ (ran 𝐹 = 𝐵 ↔ dom ^◡𝐹 = 𝐵)
5	4	anbi2i 732	. . . 4 ⊢ ((Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun ^◡𝐹 ∧ dom ^◡𝐹 = 𝐵))
6		df-fn 6052	. . . 4 ⊢ (^◡𝐹 Fn 𝐵 ↔ (Fun ^◡𝐹 ∧ dom ^◡𝐹 = 𝐵))
7	5, 6	bitr4i 267	. . 3 ⊢ ((Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ ^◡𝐹 Fn 𝐵)
8	7	anbi2i 732	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ (Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵)) ↔ (𝐹 Fn 𝐴 ∧ ^◡𝐹 Fn 𝐵))
9	1, 2, 8	3bitri 286	1 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ ^◡𝐹 Fn 𝐵))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 196 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ^◡ccnv 5265 dom cdm 5266 ran crn 5267 Fun wfun 6043 Fn wfn 6044 –1-1-onto→wf1o 6048

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-in 3722 df-ss 3729 df-rn 5277 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056

This theorem is referenced by: f1ocnv 6310 f1oun 6317 f1o00 6332 f1oi 6335 f1osn 6337 f1oprswap 6341 f1ompt 6545 f1ofveu 6808 f1ocnvd 7049 curry1 7437 curry2 7440 mapsnf1o2 8071 omxpenlem 8226 sbthlem9 8243 compssiso 9388 mptfzshft 14709 fsumrev 14710 fprodrev 14906 invf1o 16630 mhmf1o 17546 grpinvf1o 17686 ghmf1o 17891 rhmf1o 18934 srngf1o 19056 lmhmf1o 19248 hmeof1o2 21768 axcontlem2 26044 f1o3d 29740 padct 29806 f1od2 29808 cdleme51finvN 36346 fsovf1od 38812 mgmhmf1o 42297 rnghmf1o 42413