Theorem elmpom 6412

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 6033	. . . . . . . 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 4938	. . . . . 6 |- dom F = dom { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) }
5		dmoprabss 6113	. . . . . 6 |- dom { <. <. x , y >. , r >. | ( ( x e. A /\ y e. B ) /\ r = C ) } C_ ( A X. B )
6	4, 5	eqsstri 3260	. . . . 5 |- dom F C_ ( A X. B )
7		2ndexg 6340	. . . . . . 7 |- ( D e. F -> ( 2nd ` D ) e. _V )
8	1	mpofun 6133	. . . . . . . . . . 11 |- Fun F
9		funrel 5350	. . . . . . . . . . 11 |- ( Fun F -> Rel F )
10	8, 9	ax-mp 5	. . . . . . . . . 10 |- Rel F
11		1st2nd 6353	. . . . . . . . . 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 3868	. . . . . . . 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 2952	. . . . . 6 |- ( D e. F -> E. s <. ( 1st ` D ) , s >. e. F )
18		1stexg 6339	. . . . . . 7 |- ( D e. F -> ( 1st ` D ) e. _V )
19		eldm2g 4933	. . . . . . 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 3226	. . . 4 |- ( D e. F -> ( 1st ` D ) e. ( A X. B ) )
23		elex2 2820	. . . 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 5165	. . 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 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   <.cop 3676   X. cxp 4729   dom cdm 4731   Rel wrel 4736   Fun wfun 5327   ` cfv 5333   {coprab 6029   e. cmpo 6030   1stc1st 6310   2ndc2nd 6311

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313

This theorem is referenced by:  clwwlknonmpo  16352