dff1o4 - Metamath Proof Explorer

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Theorem dff1o4 6145

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 6142	. 2 ⊢ (𝐹:𝐴–1-1-onto→𝐵 ↔ (𝐹 Fn 𝐴 ∧ Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵))
2		3anass 1042	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (𝐹 Fn 𝐴 ∧ (Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵)))
3		df-rn 5125	. . . . . 6 ⊢ ran 𝐹 = dom ^◡𝐹
4	3	eqeq1i 2627	. . . . 5 ⊢ (ran 𝐹 = 𝐵 ↔ dom ^◡𝐹 = 𝐵)
5	4	anbi2i 730	. . . 4 ⊢ ((Fun ^◡𝐹 ∧ ran 𝐹 = 𝐵) ↔ (Fun ^◡𝐹 ∧ dom ^◡𝐹 = 𝐵))
6		df-fn 5891	. . . 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 730	. 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 384 ∧ w3a 1037 = wceq 1483 ^◡ccnv 5113 dom cdm 5114 ran crn 5115 Fun wfun 5882 Fn wfn 5883 –1-1-onto→wf1o 5887

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-in 3581 df-ss 3588 df-rn 5125 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895

This theorem is referenced by: f1ocnv 6149 f1oun 6156 f1o00 6171 f1oi 6174 f1osn 6176 f1oprswap 6180 f1ompt 6382 f1ofveu 6645 f1ocnvd 6884 curry1 7269 curry2 7272 mapsnf1o2 7905 omxpenlem 8061 sbthlem9 8078 compssiso 9196 mptfzshft 14510 fsumrev 14511 fprodrev 14707 invf1o 16429 mhmf1o 17345 grpinvf1o 17485 ghmf1o 17690 rhmf1o 18732 srngf1o 18854 lmhmf1o 19046 hmeof1o2 21566 axcontlem2 25845 f1o3d 29431 padct 29497 f1od2 29499 cdleme51finvN 35844 fsovf1od 38310 mgmhmf1o 41787 rnghmf1o 41903