Theorem fv1stcnv 31680

Description: The value of the converse of 1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)

Assertion

Ref	Expression
fv1stcnv	|- ( ( X e. A /\ Y e. V ) -> ( `' ( 1st |` ( A X. { Y } ) ) ` X ) = <. X , Y >. )

Proof of Theorem fv1stcnv

Step	Hyp	Ref	Expression
1		snidg 4206	. . . . 5 |- ( Y e. V -> Y e. { Y } )
2	1	anim2i 593	. . . 4 |- ( ( X e. A /\ Y e. V ) -> ( X e. A /\ Y e. { Y } ) )
3		eqid 2622	. . . 4 |- X = X
4	2, 3	jctil 560	. . 3 |- ( ( X e. A /\ Y e. V ) -> ( X = X /\ ( X e. A /\ Y e. { Y } ) ) )
5		opex 4932	. . . . . . 7 |- <. X , Y >. e. _V
6		brcnvg 5303	. . . . . . 7 |- ( ( X e. A /\ <. X , Y >. e. _V ) -> ( X `' ( 1st |` ( A X. { Y } ) ) <. X , Y >. <-> <. X , Y >. ( 1st |` ( A X. { Y } ) ) X ) )
7	5, 6	mpan2 707	. . . . . 6 |- ( X e. A -> ( X `' ( 1st |` ( A X. { Y } ) ) <. X , Y >. <-> <. X , Y >. ( 1st |` ( A X. { Y } ) ) X ) )
8		brresg 5405	. . . . . 6 |- ( X e. A -> ( <. X , Y >. ( 1st |` ( A X. { Y } ) ) X <-> ( <. X , Y >. 1st X /\ <. X , Y >. e. ( A X. { Y } ) ) ) )
9	7, 8	bitrd 268	. . . . 5 |- ( X e. A -> ( X `' ( 1st |` ( A X. { Y } ) ) <. X , Y >. <-> ( <. X , Y >. 1st X /\ <. X , Y >. e. ( A X. { Y } ) ) ) )
10	9	adantr 481	. . . 4 |- ( ( X e. A /\ Y e. V ) -> ( X `' ( 1st |` ( A X. { Y } ) ) <. X , Y >. <-> ( <. X , Y >. 1st X /\ <. X , Y >. e. ( A X. { Y } ) ) ) )
11		opelxp 5146	. . . . . 6 |- ( <. X , Y >. e. ( A X. { Y } ) <-> ( X e. A /\ Y e. { Y } ) )
12	11	anbi2i 730	. . . . 5 |- ( ( <. X , Y >. 1st X /\ <. X , Y >. e. ( A X. { Y } ) ) <-> ( <. X , Y >. 1st X /\ ( X e. A /\ Y e. { Y } ) ) )
13		br1steqg 31672	. . . . . 6 |- ( ( X e. A /\ Y e. V ) -> ( <. X , Y >. 1st X <-> X = X ) )
14	13	anbi1d 741	. . . . 5 |- ( ( X e. A /\ Y e. V ) -> ( ( <. X , Y >. 1st X /\ ( X e. A /\ Y e. { Y } ) ) <-> ( X = X /\ ( X e. A /\ Y e. { Y } ) ) ) )
15	12, 14	syl5bb 272	. . . 4 |- ( ( X e. A /\ Y e. V ) -> ( ( <. X , Y >. 1st X /\ <. X , Y >. e. ( A X. { Y } ) ) <-> ( X = X /\ ( X e. A /\ Y e. { Y } ) ) ) )
16	10, 15	bitrd 268	. . 3 |- ( ( X e. A /\ Y e. V ) -> ( X `' ( 1st |` ( A X. { Y } ) ) <. X , Y >. <-> ( X = X /\ ( X e. A /\ Y e. { Y } ) ) ) )
17	4, 16	mpbird 247	. 2 |- ( ( X e. A /\ Y e. V ) -> X `' ( 1st |` ( A X. { Y } ) ) <. X , Y >. )
18		1stconst 7265	. . . 4 |- ( Y e. V -> ( 1st |` ( A X. { Y } ) ) : ( A X. { Y } ) -1-1-onto-> A )
19		f1ocnv 6149	. . . 4 |- ( ( 1st |` ( A X. { Y } ) ) : ( A X. { Y } ) -1-1-onto-> A -> `' ( 1st |` ( A X. { Y } ) ) : A -1-1-onto-> ( A X. { Y } ) )
20		f1ofn 6138	. . . 4 |- ( `' ( 1st |` ( A X. { Y } ) ) : A -1-1-onto-> ( A X. { Y } ) -> `' ( 1st |` ( A X. { Y } ) ) Fn A )
21	18, 19, 20	3syl 18	. . 3 |- ( Y e. V -> `' ( 1st |` ( A X. { Y } ) ) Fn A )
22		simpl 473	. . 3 |- ( ( X e. A /\ Y e. V ) -> X e. A )
23		fnbrfvb 6236	. . 3 |- ( ( `' ( 1st |` ( A X. { Y } ) ) Fn A /\ X e. A ) -> ( ( `' ( 1st |` ( A X. { Y } ) ) ` X ) = <. X , Y >. <-> X `' ( 1st |` ( A X. { Y } ) ) <. X , Y >. ) )
24	21, 22, 23	syl2an2 875	. 2 |- ( ( X e. A /\ Y e. V ) -> ( ( `' ( 1st |` ( A X. { Y } ) ) ` X ) = <. X , Y >. <-> X `' ( 1st |` ( A X. { Y } ) ) <. X , Y >. ) )
25	17, 24	mpbird 247	1 |- ( ( X e. A /\ Y e. V ) -> ( `' ( 1st |` ( A X. { Y } ) ) ` X ) = <. X , Y >. )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   |` cres 5116   Fn wfn 5883   -1-1-onto->wf1o 5887   ` cfv 5888   1stc1st 7166

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-csb 3534  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-iun 4522  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by: (None)