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Theorem ovmptss 6896
 Description: If all the values of the mapping are subsets of a class , then so is any evaluation of the mapping. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
ovmptss.1
Assertion
Ref Expression
ovmptss
Distinct variable groups:   ,,   ,   ,,
Allowed substitution hints:   ()   (,)   (,)   (,)   (,)

Proof of Theorem ovmptss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovmptss.1 . . . 4
2 mpt2mptsx 6875 . . . 4
31, 2eqtri 2493 . . 3
43fvmptss 5973 . 2
5 vex 3034 . . . . . . . 8
6 vex 3034 . . . . . . . 8
75, 6op1std 6822 . . . . . . 7
87csbeq1d 3356 . . . . . 6
95, 6op2ndd 6823 . . . . . . . 8
109csbeq1d 3356 . . . . . . 7
1110csbeq2dv 3785 . . . . . 6
128, 11eqtrd 2505 . . . . 5
1312sseq1d 3445 . . . 4
1413raliunxp 4979 . . 3
15 nfcv 2612 . . . . 5
16 nfcv 2612 . . . . . 6
17 nfcsb1v 3365 . . . . . 6
1816, 17nfxp 4866 . . . . 5
19 sneq 3969 . . . . . 6
20 csbeq1a 3358 . . . . . 6
2119, 20xpeq12d 4864 . . . . 5
2215, 18, 21cbviun 4306 . . . 4
2322raleqi 2977 . . 3
24 nfv 1769 . . . 4
25 nfcsb1v 3365 . . . . . 6
26 nfcv 2612 . . . . . 6
2725, 26nfss 3411 . . . . 5
2817, 27nfral 2789 . . . 4
29 nfv 1769 . . . . . 6
30 nfcsb1v 3365 . . . . . . 7
31 nfcv 2612 . . . . . . 7
3230, 31nfss 3411 . . . . . 6
33 csbeq1a 3358 . . . . . . 7
3433sseq1d 3445 . . . . . 6
3529, 32, 34cbvral 3001 . . . . 5
36 csbeq1a 3358 . . . . . . 7
3736sseq1d 3445 . . . . . 6
3820, 37raleqbidv 2987 . . . . 5
3935, 38syl5bb 265 . . . 4
4024, 28, 39cbvral 3001 . . 3
4114, 23, 403bitr4ri 286 . 2
42 df-ov 6311 . . 3
4342sseq1i 3442 . 2
444, 41, 433imtr4i 274 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1452  wral 2756  csb 3349   wss 3390  csn 3959  cop 3965  ciun 4269   cmpt 4454   cxp 4837  cfv 5589  (class class class)co 6308   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-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813 This theorem is referenced by:  relmpt2opab  6897  relxpchom  16144  reldv  22904
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