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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.
StepHypRef 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
86, 7nfcsb 3452 . . . . 5
9 csbeq1a 3443 . . . . 5
10 csbeq1a 3443 . . . . . 6
11 csbeq1a 3443 . . . . . 6
1210, 11sylan9eqr 2520 . . . . 5
131, 2, 3, 4, 5, 8, 9, 12cbvmpt2x 6375 . . . 4
14 fmpt2x.1 . . . 4
15 vex 3112 . . . . . . . 8
16 vex 3112 . . . . . . . 8
1715, 16op1std 6810 . . . . . . 7
1817csbeq1d 3441 . . . . . 6
1915, 16op2ndd 6811 . . . . . . . 8
2019csbeq1d 3441 . . . . . . 7
2120csbeq2dv 3835 . . . . . 6
2218, 21eqtrd 2498 . . . . 5
2322mpt2mptx 6393 . . . 4
2413, 14, 233eqtr4i 2496 . . 3
2524dmmptss 5508 . 2
26 nfcv 2619 . . 3
27 nfcv 2619 . . . 4
2827, 2nfxp 5031 . . 3
29 sneq 4039 . . . 4
3029, 9xpeq12d 5029 . . 3
3126, 28, 30cbviun 4367 . 2
3225, 31sseqtr4i 3536 1
 Colors of variables: wff setvar class 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
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