Theorem dfmpt2 6890

Description: Alternate definition for the "maps to" notation df-mpt2 6301 (although it requires that be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
dfmpt2.1

Assertion

Ref	Expression
dfmpt2

Distinct variable groups:   ,,   ,,

Proof of Theorem dfmpt2

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpt2mpts 6864	. 2
2		dfmpt2.1	. . . . 5
3	2	csbex 4585	. . . 4
4	3	csbex 4585	. . 3
5	4	dfmpt 6076	. 2
6		nfcv 2619	. . . . 5
7		nfcsb1v 3450	. . . . 5
8	6, 7	nfop 4233	. . . 4
9	8	nfsn 4087	. . 3
10		nfcv 2619	. . . . 5
11		nfcv 2619	. . . . . 6
12		nfcsb1v 3450	. . . . . 6
13	11, 12	nfcsb 3452	. . . . 5
14	10, 13	nfop 4233	. . . 4
15	14	nfsn 4087	. . 3
16		nfcv 2619	. . 3
17		id 22	. . . . 5
18		csbopeq1a 6853	. . . . 5
19	17, 18	opeq12d 4225	. . . 4
20	19	sneqd 4041	. . 3
21	9, 15, 16, 20	iunxpf 5156	. 2
22	1, 5, 21	3eqtri 2490	1

Syntax hints:  =wceq 1395  e.wcel 1818  cvv 3109  [_csb 3434  {csn 4029  <.cop 4035  U_ciun 4330  e.cmpt 4510  X.cxp 5002  `cfv 5593  e.cmpt2 6298  c1st 6798  c2nd 6799

This theorem is referenced by:  fpar  6904

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-8 1820  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-pow 4630  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-fal 1401  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-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  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-uni 4250  df-iun 4332  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-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801