f1ocnv2d - Metamath Proof Explorer

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Theorem f1ocnv2d 6526

Description: Describe an implicit one-to-one onto function. (Contributed by Mario Carneiro, 30-Apr-2015.)

**Assertion**
Ref	Expression
f1ocnv2d

Distinct variable groups: ,, ,, , , ,,

**Proof of Theorem f1ocnv2d**
Step	Hyp	Ref	Expression
1		f1od.1	. 2
2		f1o2d.2	. 2
3		f1o2d.3	. 2
4		eleq1a 2540	. . . . . 6
5	2, 4	syl 16	. . . . 5
6	5	impr 619	. . . 4
7		f1o2d.4	. . . . . . . 8
8	7	biimpar 485	. . . . . . 7
9	8	exp42 611	. . . . . 6
10	9	com34 83	. . . . 5
11	10	imp32 433	. . . 4
12	6, 11	jcai 536	. . 3
13		eleq1a 2540	. . . . . 6
14	3, 13	syl 16	. . . . 5
15	14	impr 619	. . . 4
16	7	biimpa 484	. . . . . . . 8
17	16	exp42 611	. . . . . . 7
18	17	com23 78	. . . . . 6
19	18	com34 83	. . . . 5
20	19	imp32 433	. . . 4
21	15, 20	jcai 536	. . 3
22	12, 21	impbida 832	. 2
23	1, 2, 3, 22	f1ocnvd 6524	1

Colors of variables: wff setvar class

Syntax hints: ->wi 4 <->wb 184 /\wa 369 =wceq 1395 e.wcel 1818 e.cmpt 4510 `'ccnv 5003 -1-1-onto->wf1o 5592

This theorem is referenced by: f1o2d 6527 negiso 10544 iccf1o 11693 bitsf1ocnv 14094 grpinvcnv 16106 grplactcnv 16138 issrngd 17510 opncldf1 19585 txhmeo 20304 ptuncnv 20308 icopnfcnv 21442 iccpnfcnv 21444 xrge0iifcnv 27915

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pr 4691

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-rab 2816 df-v 3111 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-sn 4030 df-pr 4032 df-op 4036 df-br 4453 df-opab 4511 df-mpt 4512 df-id 4800 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-fun 5595 df-fn 5596 df-f 5597 df-f1 5598 df-fo 5599 df-f1o 5600