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Theorem fvmptss2 5275
 Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
Hypotheses
Ref Expression
fvmptss2.1
fvmptss2.2
Assertion
Ref Expression
fvmptss2
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem fvmptss2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fvss 5217 . 2
2 fvmptss2.2 . . . . . 6
32funmpt2 4967 . . . . 5
4 funrel 4947 . . . . 5
53, 4ax-mp 7 . . . 4
65brrelexi 4412 . . 3
7 nfcv 2194 . . . 4
8 nfmpt1 3878 . . . . . . 7
92, 8nfcxfr 2191 . . . . . 6
10 nfcv 2194 . . . . . 6
117, 9, 10nfbr 3836 . . . . 5
12 nfv 1437 . . . . 5
1311, 12nfim 1480 . . . 4
14 breq1 3795 . . . . 5
15 fvmptss2.1 . . . . . 6
1615sseq2d 3001 . . . . 5
1714, 16imbi12d 227 . . . 4
18 df-br 3793 . . . . 5
19 opabid 4022 . . . . . . 7
20 eqimss 3025 . . . . . . . 8
2120adantl 266 . . . . . . 7
2219, 21sylbi 118 . . . . . 6
23 df-mpt 3848 . . . . . . 7
242, 23eqtri 2076 . . . . . 6
2522, 24eleq2s 2148 . . . . 5
2618, 25sylbi 118 . . . 4
277, 13, 17, 26vtoclgf 2629 . . 3
286, 27mpcom 36 . 2
291, 28mpg 1356 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 101   wceq 1259   wcel 1409  cvv 2574   wss 2945  cop 3406   class class class wbr 3792  copab 3845   cmpt 3846   wrel 4378   wfun 4924  cfv 4930 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-iota 4895  df-fun 4932  df-fv 4938 This theorem is referenced by:  mptfvex  5284
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