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

Step	Hyp	Ref	Expression
1		ne0i 3790	. . 3
2		ndmfv 5895	. . . 4
3	2	necon1ai 2688	. . 3
4		elfvmptrab1.f	. . . . . . . 8
5	4	dmmptss 5508	. . . . . . 7
6	5	sseli 3499	. . . . . 6
7		elfvmptrab1.v	. . . . . . 7
8		rabexg 4602	. . . . . . 7
9	6, 7, 8	3syl 20	. . . . . 6
10		nfcv 2619	. . . . . . 7
11		nfsbc1v 3347	. . . . . . . 8
12		nfcv 2619	. . . . . . . . 9
13	10, 12	nfcsb 3452	. . . . . . . 8
14	11, 13	nfrab 3039	. . . . . . 7
15		csbeq1 3437	. . . . . . . 8
16		sbceq1a 3338	. . . . . . . 8
17	15, 16	rabeqbidv 3104	. . . . . . 7
18	10, 14, 17, 4	fvmptf 5972	. . . . . 6
19	6, 9, 18	syl2anc 661	. . . . 5
20	19	eleq2d 2527	. . . 4
21		elrabi 3254	. . . . . 6
22	6, 21	anim12i 566	. . . . 5
23	22	ex 434	. . . 4
24	20, 23	sylbid 215	. . 3
25	1, 3, 24	3syl 20	. 2
26	25	pm2.43i 47	1

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