Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmmpt2ssx Structured version   Unicode version

Theorem dmmpt2ssx 6417
 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:   ,,   ,
Allowed substitution hints:   ()   (,)   (,)

Proof of Theorem dmmpt2ssx
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2573 . . . . 5
2 nfcsb1v 3284 . . . . 5
3 nfcv 2573 . . . . 5
4 nfcv 2573 . . . . 5
5 nfcsb1v 3284 . . . . 5
6 nfcv 2573 . . . . . 6
7 nfcsb1v 3284 . . . . . 6
86, 7nfcsb 3286 . . . . 5
9 csbeq1a 3260 . . . . 5
10 csbeq1a 3260 . . . . . 6
11 csbeq1a 3260 . . . . . 6
1210, 11sylan9eqr 2491 . . . . 5
131, 2, 3, 4, 5, 8, 9, 12cbvmpt2x 6151 . . . 4
14 fmpt2x.1 . . . 4
15 vex 2960 . . . . . . . 8
16 vex 2960 . . . . . . . 8
1715, 16op1std 6358 . . . . . . 7
1817csbeq1d 3258 . . . . . 6
1915, 16op2ndd 6359 . . . . . . . 8
2019csbeq1d 3258 . . . . . . 7
2120csbeq2dv 3277 . . . . . 6
2218, 21eqtrd 2469 . . . . 5
2322mpt2mptx 6165 . . . 4
2413, 14, 233eqtr4i 2467 . . 3
2524dmmptss 5367 . 2
26 nfcv 2573 . . 3
27 nfcv 2573 . . . 4
2827, 2nfxp 4905 . . 3
29 sneq 3826 . . . 4
3029, 9xpeq12d 4904 . . 3
3126, 28, 30cbviun 4129 . 2
3225, 31sseqtr4i 3382 1
 Colors of variables: wff set class Syntax hints:   wceq 1653  csb 3252   wss 3321  csn 3815  cop 3818  ciun 4094   cmpt 4267   cxp 4877   cdm 4879  cfv 5455   cmpt2 6084  c1st 6348  c2nd 6349 This theorem is referenced by:  mpt2exxg  6423  mpt2xopn0yelv  6465  mpt2xopxnop0  6467  dmcoass  14222  dvbsss  19790  perfdvf  19791 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fv 5463  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351
 Copyright terms: Public domain W3C validator