Theorem elmpom 6402

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 6022	. . . . . . . 8 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) }
3	1, 2	eqtri 2252	. . . . . . 7 |- F = { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) }
4	3	dmeqi 4932	. . . . . 6 |- dom F = dom { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) }
5		dmoprabss 6102	. . . . . 6 |- dom { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) } C_ ( A X. B )
6	4, 5	eqsstri 3259	. . . . 5 |- dom F C_ ( A X. B )
7		2ndexg 6330	. . . . . . 7 |- ( D e. F -> ( 2nd ` D ) e. _V )
8	1	mpofun 6122	. . . . . . . . . . 11 |- Fun F
9		funrel 5343	. . . . . . . . . . 11 |- ( Fun F -> Rel F )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- Rel F
11		1st2nd 6343	. . . . . . . . . 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 2300	. . . . . . . 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 3863	. . . . . . . 8 |- ( s = ( 2nd ` D ) -> <. ( 1st ` D ) , s >. = <. ( 1st ` D ) , ( 2nd ` D ) >. )
16	15	eleq1d 2300	. . . . . . 7 |- ( s = ( 2nd ` D ) -> ( <. ( 1st ` D ) , s >. e. F <-> <. ( 1st ` D ) , ( 2nd ` D ) >. e. F ) )
17	7, 14, 16	elabd 2951	. . . . . 6 |- ( D e. F -> E. s <. ( 1st ` D ) , s >. e. F )
18		1stexg 6329	. . . . . . 7 |- ( D e. F -> ( 1st ` D ) e. _V )
19		eldm2g 4927	. . . . . . 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 3225	. . . 4 |- ( D e. F -> ( 1st ` D ) e. ( A X. B ) )
23		elex2 2819	. . . 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 5158	. . 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 1397   E.wex 1540   e. wcel 2202   _Vcvv 2802   <.cop 3672   X. cxp 4723   dom cdm 4725   Rel wrel 4730   Fun wfun 5320   ` cfv 5326   {coprab 6018   e. cmpo 6019   1stc1st 6300   2ndc2nd 6301

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303

This theorem is referenced by:  clwwlknonmpo  16278