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Theorem fvmptss2 5542
 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 5481 . 2
2 fvmptss2.2 . . . . . 6
32funmpt2 5208 . . . . 5
4 funrel 5186 . . . . 5
53, 4ax-mp 5 . . . 4
65brrelex1i 4628 . . 3
7 nfcv 2299 . . . 4
8 nfmpt1 4057 . . . . . . 7
92, 8nfcxfr 2296 . . . . . 6
10 nfcv 2299 . . . . . 6
117, 9, 10nfbr 4010 . . . . 5
12 nfv 1508 . . . . 5
1311, 12nfim 1552 . . . 4
14 breq1 3968 . . . . 5
15 fvmptss2.1 . . . . . 6
1615sseq2d 3158 . . . . 5
1714, 16imbi12d 233 . . . 4
18 df-br 3966 . . . . 5
19 opabid 4217 . . . . . . 7
20 eqimss 3182 . . . . . . . 8
2120adantl 275 . . . . . . 7
2219, 21sylbi 120 . . . . . 6
23 df-mpt 4027 . . . . . . 7
242, 23eqtri 2178 . . . . . 6
2522, 24eleq2s 2252 . . . . 5
2618, 25sylbi 120 . . . 4
277, 13, 17, 26vtoclgf 2770 . . 3
286, 27mpcom 36 . 2
291, 28mpg 1431 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1335   wcel 2128  cvv 2712   wss 3102  cop 3563   class class class wbr 3965  copab 4024   cmpt 4025   wrel 4590   wfun 5163  cfv 5169 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-iota 5134  df-fun 5171  df-fv 5177 This theorem is referenced by:  mptfvex  5552
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