Theorem snelmap 39254

Description: Membership of the element in the range of a constant map. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
snelmap.a	|- ( ph -> A e. V )
snelmap.b	|- ( ph -> B e. W )
snelmap.n	|- ( ph -> A =/= (/) )
snelmap.e	|- ( ph -> ( A X. { x } ) e. ( B ^m A ) )

Assertion

Ref	Expression
snelmap	|- ( ph -> x e. B )

Proof of Theorem snelmap

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snelmap.n	. . 3 |- ( ph -> A =/= (/) )
2		n0 3931	. . 3 |- ( A =/= (/) <-> E. y y e. A )
3	1, 2	sylib 208	. 2 |- ( ph -> E. y y e. A )
4		vex 3203	. . . . . . . 8 |- x e. _V
5	4	fvconst2 6469	. . . . . . 7 |- ( y e. A -> ( ( A X. { x } ) ` y ) = x )
6	5	eqcomd 2628	. . . . . 6 |- ( y e. A -> x = ( ( A X. { x } ) ` y ) )
7	6	adantl 482	. . . . 5 |- ( ( ph /\ y e. A ) -> x = ( ( A X. { x } ) ` y ) )
8		snelmap.e	. . . . . . . 8 |- ( ph -> ( A X. { x } ) e. ( B ^m A ) )
9		snelmap.b	. . . . . . . . 9 |- ( ph -> B e. W )
10		snelmap.a	. . . . . . . . 9 |- ( ph -> A e. V )
11		elmapg 7870	. . . . . . . . 9 |- ( ( B e. W /\ A e. V ) -> ( ( A X. { x } ) e. ( B ^m A ) <-> ( A X. { x } ) : A --> B ) )
12	9, 10, 11	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( A X. { x } ) e. ( B ^m A ) <-> ( A X. { x } ) : A --> B ) )
13	8, 12	mpbid 222	. . . . . . 7 |- ( ph -> ( A X. { x } ) : A --> B )
14	13	adantr 481	. . . . . 6 |- ( ( ph /\ y e. A ) -> ( A X. { x } ) : A --> B )
15		simpr 477	. . . . . 6 |- ( ( ph /\ y e. A ) -> y e. A )
16	14, 15	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ y e. A ) -> ( ( A X. { x } ) ` y ) e. B )
17	7, 16	eqeltrd 2701	. . . 4 |- ( ( ph /\ y e. A ) -> x e. B )
18	17	ex 450	. . 3 |- ( ph -> ( y e. A -> x e. B ) )
19	18	exlimdv 1861	. 2 |- ( ph -> ( E. y y e. A -> x e. B ) )
20	3, 19	mpd 15	1 |- ( ph -> x e. B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   (/)c0 3915   {csn 4177   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859

This theorem is referenced by:  mapssbi  39405