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Theorem dmmpt2ssx2 40626
 Description: The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpt2ssx 6877. (Contributed by AV, 30-Mar-2019.)
Hypothesis
Ref Expression
dmmpt2ssx2.1
Assertion
Ref Expression
dmmpt2ssx2
Distinct variable groups:   ,   ,,
Allowed substitution hints:   ()   (,)   (,)

Proof of Theorem dmmpt2ssx2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2612 . . . . 5
2 nfcsb1v 3365 . . . . 5
3 nfcv 2612 . . . . 5
4 nfcv 2612 . . . . 5
5 nfcv 2612 . . . . . 6
6 nfcsb1v 3365 . . . . . 6
75, 6nfcsb 3367 . . . . 5
8 nfcsb1v 3365 . . . . 5
9 csbeq1a 3358 . . . . 5
10 csbeq1a 3358 . . . . . 6
11 csbeq1a 3358 . . . . . 6
1210, 11sylan9eqr 2527 . . . . 5
131, 2, 3, 4, 7, 8, 9, 12cbvmpt2x2 40625 . . . 4
14 dmmpt2ssx2.1 . . . 4
15 vex 3034 . . . . . . . 8
16 vex 3034 . . . . . . . 8
1715, 16op2ndd 6823 . . . . . . 7
1817csbeq1d 3356 . . . . . 6
1915, 16op1std 6822 . . . . . . . 8
2019csbeq1d 3356 . . . . . . 7
2120csbeq2dv 3785 . . . . . 6
2218, 21eqtrd 2505 . . . . 5
2322mpt2mptx2 40624 . . . 4
2413, 14, 233eqtr4i 2503 . . 3
2524dmmptss 5338 . 2
26 nfcv 2612 . . 3
27 nfcv 2612 . . . 4
282, 27nfxp 4866 . . 3
29 sneq 3969 . . . 4
309, 29xpeq12d 4864 . . 3
3126, 28, 30cbviun 4306 . 2
3225, 31sseqtr4i 3451 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1452  csb 3349   wss 3390  csn 3959  cop 3965  ciun 4269   cmpt 4454   cxp 4837   cdm 4839  cfv 5589   cmpt2 6310  c1st 6810  c2nd 6811 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813 This theorem is referenced by:  mpt2exxg2  40627
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