Theorem elmpom 6478

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 6093	. . . . . . . 8 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) }
3	1, 2	eqtri 2259	. . . . . . 7 |- F = { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) }
4	3	dmeqi 4984	. . . . . 6 |- dom F = dom { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) }
5		dmoprabss 6173	. . . . . 6 |- dom { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) } C_ ( A X. B )
6	4, 5	eqsstri 3280	. . . . 5 |- dom F C_ ( A X. B )
7		2ndexg 6406	. . . . . . 7 |- ( D e. F -> ( 2nd ` D ) e. _V )
8	1	mpofun 6193	. . . . . . . . . . 11 |- Fun F
9		funrel 5397	. . . . . . . . . . 11 |- ( Fun F -> Rel F )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- Rel F
11		1st2nd 6419	. . . . . . . . . 10 |- ( ( Rel F /\ D e. F ) -> D = <. ( 1st ` D ) , ( 2nd ` D ) >. )
12	10, 11	mpan 428	. . . . . . . . 9 |- ( D e. F -> D = <. ( 1st ` D ) , ( 2nd ` D ) >. )
13	12	eleq1d 2307	. . . . . . . 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 3905	. . . . . . . 8 |- ( s = ( 2nd ` D ) -> <. ( 1st ` D ) , s >. = <. ( 1st ` D ) , ( 2nd ` D ) >. )
16	15	eleq1d 2307	. . . . . . 7 |- ( s = ( 2nd ` D ) -> ( <. ( 1st ` D ) , s >. e. F <-> <. ( 1st ` D ) , ( 2nd ` D ) >. e. F ) )
17	7, 14, 16	elabd 2971	. . . . . 6 |- ( D e. F -> E. s <. ( 1st ` D ) , s >. e. F )
18		1stexg 6405	. . . . . . 7 |- ( D e. F -> ( 1st ` D ) e. _V )
19		eldm2g 4979	. . . . . . 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 3246	. . . 4 |- ( D e. F -> ( 1st ` D ) e. ( A X. B ) )
23		elex2 2838	. . . 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 5212	. . 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   <.cop 3712   X. cxp 4774   dom cdm 4776   Rel wrel 4781   Fun wfun 5374   ` cfv 5380   {coprab 6089   e. cmpo 6090   1stc1st 6376   2ndc2nd 6377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fo 5386  df-fv 5388  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379

This theorem is used by:  clwwlknonmpo  16874