Theorem 1stpreimas 29483

Description: The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)

Assertion

Ref	Expression
1stpreimas	|- ( ( Rel A /\ X e. V ) -> ( `' ( 1st |` A ) " { X } ) = ( { X } X. ( A " { X } ) ) )

Proof of Theorem 1stpreimas

Dummy variables x z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1st2ndb 7206	. . . . . . . . 9 |- ( z e. ( _V X. _V ) <-> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
2	1	biimpi 206	. . . . . . . 8 |- ( z e. ( _V X. _V ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
3	2	ad2antrl 764	. . . . . . 7 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
4		fvex 6201	. . . . . . . . . . . 12 |- ( 1st ` z ) e. _V
5	4	elsn 4192	. . . . . . . . . . 11 |- ( ( 1st ` z ) e. { X } <-> ( 1st ` z ) = X )
6	5	biimpi 206	. . . . . . . . . 10 |- ( ( 1st ` z ) e. { X } -> ( 1st ` z ) = X )
7	6	ad2antrl 764	. . . . . . . . 9 |- ( ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) -> ( 1st ` z ) = X )
8	7	adantl 482	. . . . . . . 8 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) -> ( 1st ` z ) = X )
9	8	opeq1d 4408	. . . . . . 7 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. X , ( 2nd ` z ) >. )
10	3, 9	eqtrd 2656	. . . . . 6 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) -> z = <. X , ( 2nd ` z ) >. )
11		simplr 792	. . . . . . 7 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) -> X e. V )
12		simprrr 805	. . . . . . 7 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) -> ( 2nd ` z ) e. ( A " { X } ) )
13		elimasng 5491	. . . . . . . 8 |- ( ( X e. V /\ ( 2nd ` z ) e. ( A " { X } ) ) -> ( ( 2nd ` z ) e. ( A " { X } ) <-> <. X , ( 2nd ` z ) >. e. A ) )
14	13	biimpa 501	. . . . . . 7 |- ( ( ( X e. V /\ ( 2nd ` z ) e. ( A " { X } ) ) /\ ( 2nd ` z ) e. ( A " { X } ) ) -> <. X , ( 2nd ` z ) >. e. A )
15	11, 12, 12, 14	syl21anc 1325	. . . . . 6 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) -> <. X , ( 2nd ` z ) >. e. A )
16	10, 15	eqeltrd 2701	. . . . 5 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) -> z e. A )
17		fvres 6207	. . . . . . 7 |- ( z e. A -> ( ( 1st |` A ) ` z ) = ( 1st ` z ) )
18	16, 17	syl 17	. . . . . 6 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) -> ( ( 1st |` A ) ` z ) = ( 1st ` z ) )
19	18, 8	eqtrd 2656	. . . . 5 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) -> ( ( 1st |` A ) ` z ) = X )
20	16, 19	jca 554	. . . 4 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) -> ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) )
21		df-rel 5121	. . . . . . . . 9 |- ( Rel A <-> A C_ ( _V X. _V ) )
22	21	biimpi 206	. . . . . . . 8 |- ( Rel A -> A C_ ( _V X. _V ) )
23	22	adantr 481	. . . . . . 7 |- ( ( Rel A /\ X e. V ) -> A C_ ( _V X. _V ) )
24	23	sselda 3603	. . . . . 6 |- ( ( ( Rel A /\ X e. V ) /\ z e. A ) -> z e. ( _V X. _V ) )
25	24	adantrr 753	. . . . 5 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> z e. ( _V X. _V ) )
26	17	ad2antrl 764	. . . . . . . 8 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> ( ( 1st |` A ) ` z ) = ( 1st ` z ) )
27		simprr 796	. . . . . . . 8 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> ( ( 1st |` A ) ` z ) = X )
28	26, 27	eqtr3d 2658	. . . . . . 7 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> ( 1st ` z ) = X )
29	28, 5	sylibr 224	. . . . . 6 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> ( 1st ` z ) e. { X } )
30	28, 29	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> X e. { X } )
31		simpr 477	. . . . . . . . . . 11 |- ( ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) /\ x = X ) -> x = X )
32	31	opeq1d 4408	. . . . . . . . . 10 |- ( ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) /\ x = X ) -> <. x , ( 2nd ` z ) >. = <. X , ( 2nd ` z ) >. )
33	32	eleq1d 2686	. . . . . . . . 9 |- ( ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) /\ x = X ) -> ( <. x , ( 2nd ` z ) >. e. A <-> <. X , ( 2nd ` z ) >. e. A ) )
34		1st2nd 7214	. . . . . . . . . . . 12 |- ( ( Rel A /\ z e. A ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
35	34	ad2ant2r 783	. . . . . . . . . . 11 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
36	28	opeq1d 4408	. . . . . . . . . . 11 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. = <. X , ( 2nd ` z ) >. )
37	35, 36	eqtrd 2656	. . . . . . . . . 10 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> z = <. X , ( 2nd ` z ) >. )
38		simprl 794	. . . . . . . . . 10 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> z e. A )
39	37, 38	eqeltrrd 2702	. . . . . . . . 9 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> <. X , ( 2nd ` z ) >. e. A )
40	30, 33, 39	rspcedvd 3317	. . . . . . . 8 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> E. x e. { X } <. x , ( 2nd ` z ) >. e. A )
41		df-rex 2918	. . . . . . . 8 |- ( E. x e. { X } <. x , ( 2nd ` z ) >. e. A <-> E. x ( x e. { X } /\ <. x , ( 2nd ` z ) >. e. A ) )
42	40, 41	sylib 208	. . . . . . 7 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> E. x ( x e. { X } /\ <. x , ( 2nd ` z ) >. e. A ) )
43		fvex 6201	. . . . . . . 8 |- ( 2nd ` z ) e. _V
44	43	elima3 5473	. . . . . . 7 |- ( ( 2nd ` z ) e. ( A " { X } ) <-> E. x ( x e. { X } /\ <. x , ( 2nd ` z ) >. e. A ) )
45	42, 44	sylibr 224	. . . . . 6 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> ( 2nd ` z ) e. ( A " { X } ) )
46	29, 45	jca 554	. . . . 5 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) )
47	25, 46	jca 554	. . . 4 |- ( ( ( Rel A /\ X e. V ) /\ ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) -> ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) )
48	20, 47	impbida 877	. . 3 |- ( ( Rel A /\ X e. V ) -> ( ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) <-> ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) )
49		elxp7 7201	. . . 4 |- ( z e. ( { X } X. ( A " { X } ) ) <-> ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) )
50	49	a1i 11	. . 3 |- ( ( Rel A /\ X e. V ) -> ( z e. ( { X } X. ( A " { X } ) ) <-> ( z e. ( _V X. _V ) /\ ( ( 1st ` z ) e. { X } /\ ( 2nd ` z ) e. ( A " { X } ) ) ) ) )
51		fo1st 7188	. . . . . . 7 |- 1st : _V -onto-> _V
52		fofn 6117	. . . . . . 7 |- ( 1st : _V -onto-> _V -> 1st Fn _V )
53	51, 52	ax-mp 5	. . . . . 6 |- 1st Fn _V
54		ssv 3625	. . . . . 6 |- A C_ _V
55		fnssres 6004	. . . . . 6 |- ( ( 1st Fn _V /\ A C_ _V ) -> ( 1st |` A ) Fn A )
56	53, 54, 55	mp2an 708	. . . . 5 |- ( 1st |` A ) Fn A
57		fniniseg 6338	. . . . 5 |- ( ( 1st |` A ) Fn A -> ( z e. ( `' ( 1st |` A ) " { X } ) <-> ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) )
58	56, 57	ax-mp 5	. . . 4 |- ( z e. ( `' ( 1st |` A ) " { X } ) <-> ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) )
59	58	a1i 11	. . 3 |- ( ( Rel A /\ X e. V ) -> ( z e. ( `' ( 1st |` A ) " { X } ) <-> ( z e. A /\ ( ( 1st |` A ) ` z ) = X ) ) )
60	48, 50, 59	3bitr4rd 301	. 2 |- ( ( Rel A /\ X e. V ) -> ( z e. ( `' ( 1st |` A ) " { X } ) <-> z e. ( { X } X. ( A " { X } ) ) ) )
61	60	eqrdv 2620	1 |- ( ( Rel A /\ X e. V ) -> ( `' ( 1st |` A ) " { X } ) = ( { X } X. ( A " { X } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   _Vcvv 3200   C_ wss 3574   {csn 4177   <.cop 4183   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   Rel wrel 5119   Fn wfn 5883   -onto->wfo 5886   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

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-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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  gsummpt2d  29781