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Theorem elfvmptrab1 5976
Description: Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypotheses
Ref Expression
elfvmptrab1.f
elfvmptrab1.v
Assertion
Ref Expression
elfvmptrab1
Distinct variable groups:   ,M,   ,   , ,   ,   ,

Proof of Theorem elfvmptrab1
StepHypRef Expression
1 ne0i 3790 . . 3
2 ndmfv 5895 . . . 4
32necon1ai 2688 . . 3
4 elfvmptrab1.f . . . . . . . 8
54dmmptss 5508 . . . . . . 7
65sseli 3499 . . . . . 6
7 elfvmptrab1.v . . . . . . 7
8 rabexg 4602 . . . . . . 7
96, 7, 83syl 20 . . . . . 6
10 nfcv 2619 . . . . . . 7
11 nfsbc1v 3347 . . . . . . . 8
12 nfcv 2619 . . . . . . . . 9
1310, 12nfcsb 3452 . . . . . . . 8
1411, 13nfrab 3039 . . . . . . 7
15 csbeq1 3437 . . . . . . . 8
16 sbceq1a 3338 . . . . . . . 8
1715, 16rabeqbidv 3104 . . . . . . 7
1810, 14, 17, 4fvmptf 5972 . . . . . 6
196, 9, 18syl2anc 661 . . . . 5
2019eleq2d 2527 . . . 4
21 elrabi 3254 . . . . . 6
226, 21anim12i 566 . . . . 5
2322ex 434 . . . 4
2420, 23sylbid 215 . . 3
251, 3, 243syl 20 . 2
2625pm2.43i 47 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  {crab 2811   cvv 3109  [.wsbc 3327  [_csb 3434   c0 3784  e.cmpt 4510  domcdm 5004  `cfv 5593
This theorem is referenced by:  elfvmptrab  5977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fv 5601
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