Theorem dmmpt2ssx 6865

Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)

Hypothesis

Ref	Expression
fmpt2x.1

Assertion

Ref	Expression
dmmpt2ssx

Distinct variable groups:   ,,   ,

Proof of Theorem dmmpt2ssx

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2619	. . . . 5
2		nfcsb1v 3450	. . . . 5
3		nfcv 2619	. . . . 5
4		nfcv 2619	. . . . 5
5		nfcsb1v 3450	. . . . 5
6		nfcv 2619	. . . . . 6
7		nfcsb1v 3450	. . . . . 6
8	6, 7	nfcsb 3452	. . . . 5
9		csbeq1a 3443	. . . . 5
10		csbeq1a 3443	. . . . . 6
11		csbeq1a 3443	. . . . . 6
12	10, 11	sylan9eqr 2520	. . . . 5
13	1, 2, 3, 4, 5, 8, 9, 12	cbvmpt2x 6375	. . . 4
14		fmpt2x.1	. . . 4
15		vex 3112	. . . . . . . 8
16		vex 3112	. . . . . . . 8
17	15, 16	op1std 6810	. . . . . . 7
18	17	csbeq1d 3441	. . . . . 6
19	15, 16	op2ndd 6811	. . . . . . . 8
20	19	csbeq1d 3441	. . . . . . 7
21	20	csbeq2dv 3835	. . . . . 6
22	18, 21	eqtrd 2498	. . . . 5
23	22	mpt2mptx 6393	. . . 4
24	13, 14, 23	3eqtr4i 2496	. . 3
25	24	dmmptss 5508	. 2
26		nfcv 2619	. . 3
27		nfcv 2619	. . . 4
28	27, 2	nfxp 5031	. . 3
29		sneq 4039	. . . 4
30	29, 9	xpeq12d 5029	. . 3
31	26, 28, 30	cbviun 4367	. 2
32	25, 31	sseqtr4i 3536	1

Syntax hints:  =wceq 1395  [_csb 3434  C_wss 3475  {csn 4029  <.cop 4035  U_ciun 4330  e.cmpt 4510  X.cxp 5002  domcdm 5004  `cfv 5593  e.cmpt2 6298  c1st 6798  c2nd 6799

This theorem is referenced by:  mpt2exxg  6874  mpt2xopn0yelv  6960  mpt2xopxnop0  6962  dmcoass  15393  ply1frcl  18355  dvbsss  22306  perfdvf  22307

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-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-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-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fv 5601  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801