Theorem elmpom 6396

Description: If a maps-to operation is inhabited, the first class it is defined with is inhabited. (Contributed by Jim Kingdon, 4-Mar-2026.)

Hypothesis

Ref	Expression
elmpoex.f	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
elmpom	|- ( D e. F -> E. z z e. A )

Distinct variable groups:   x, A, y   z, A   x, B, y

Allowed substitution hints:   B( z)   C( x, y, z)   D( x, y, z)   F( x, y, z)

Proof of Theorem elmpom

Dummy variables r w s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmpoex.f	. . . . . . . 8 |- F = ( x e. A , y e. B |-> C )
2		df-mpo 6016	. . . . . . . 8 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) }
3	1, 2	eqtri 2250	. . . . . . 7 |- F = { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) }
4	3	dmeqi 4928	. . . . . 6 |- dom F = dom { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) }
5		dmoprabss 6096	. . . . . 6 |- dom { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) } C_ ( A X. B )
6	4, 5	eqsstri 3257	. . . . 5 |- dom F C_ ( A X. B )
7		2ndexg 6324	. . . . . . 7 |- ( D e. F -> ( 2nd ` D ) e. _V )
8	1	mpofun 6116	. . . . . . . . . . 11 |- Fun F
9		funrel 5339	. . . . . . . . . . 11 |- ( Fun F -> Rel F )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- Rel F
11		1st2nd 6337	. . . . . . . . . 10 |- ( ( Rel F /\ D e. F ) -> D = <. ( 1st ` D ) , ( 2nd ` D ) >. )
12	10, 11	mpan 424	. . . . . . . . 9 |- ( D e. F -> D = <. ( 1st ` D ) , ( 2nd ` D ) >. )
13	12	eleq1d 2298	. . . . . . . 8 |- ( D e. F -> ( D e. F <-> <. ( 1st ` D ) , ( 2nd ` D ) >. e. F ) )
14	13	ibi 176	. . . . . . 7 |- ( D e. F -> <. ( 1st ` D ) , ( 2nd ` D ) >. e. F )
15		opeq2 3859	. . . . . . . 8 |- ( s = ( 2nd ` D ) -> <. ( 1st ` D ) , s >. = <. ( 1st ` D ) , ( 2nd ` D ) >. )
16	15	eleq1d 2298	. . . . . . 7 |- ( s = ( 2nd ` D ) -> ( <. ( 1st ` D ) , s >. e. F <-> <. ( 1st ` D ) , ( 2nd ` D ) >. e. F ) )
17	7, 14, 16	elabd 2949	. . . . . 6 |- ( D e. F -> E. s <. ( 1st ` D ) , s >. e. F )
18		1stexg 6323	. . . . . . 7 |- ( D e. F -> ( 1st ` D ) e. _V )
19		eldm2g 4923	. . . . . . 7 |- ( ( 1st ` D ) e. _V -> ( ( 1st ` D ) e. dom F <-> E. s <. ( 1st ` D ) , s >. e. F ) )
20	18, 19	syl 14	. . . . . 6 |- ( D e. F -> ( ( 1st ` D ) e. dom F <-> E. s <. ( 1st ` D ) , s >. e. F ) )
21	17, 20	mpbird 167	. . . . 5 |- ( D e. F -> ( 1st ` D ) e. dom F )
22	6, 21	sselid 3223	. . . 4 |- ( D e. F -> ( 1st ` D ) e. ( A X. B ) )
23		elex2 2817	. . . 4 |- ( ( 1st ` D ) e. ( A X. B ) -> E. r r e. ( A X. B ) )
24	22, 23	syl 14	. . 3 |- ( D e. F -> E. r r e. ( A X. B ) )
25		xpm 5154	. . 3 |- ( ( E. z z e. A /\ E. w w e. B ) <-> E. r r e. ( A X. B ) )
26	24, 25	sylibr 134	. 2 |- ( D e. F -> ( E. z z e. A /\ E. w w e. B ) )
27	26	simpld 112	1 |- ( D e. F -> E. z z e. A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2800   <.cop 3670   X. cxp 4719   dom cdm 4721   Rel wrel 4726   Fun wfun 5316   ` cfv 5322   {coprab 6012   e. cmpo 6013   1stc1st 6294   2ndc2nd 6295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4203  ax-pow 4260  ax-pr 4295  ax-un 4526

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3890  df-br 4085  df-opab 4147  df-mpt 4148  df-id 4386  df-xp 4727  df-rel 4728  df-cnv 4729  df-co 4730  df-dm 4731  df-rn 4732  df-iota 5282  df-fun 5324  df-fn 5325  df-f 5326  df-fo 5328  df-fv 5330  df-oprab 6015  df-mpo 6016  df-1st 6296  df-2nd 6297

This theorem is referenced by:  clwwlknonmpo  16213