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Theorem fv2ndcnv 31681
Description: The value of the converse of 2nd restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
Assertion
Ref Expression
fv2ndcnv |- ( ( X e. V /\ Y e. A ) -> ( `' ( 2nd |` ( { X } X. A ) ) ` Y ) = <. X , Y >. )
Proof of Theorem fv2ndcnv
StepHypRef Expression
1 snidg 4206 . . . 4 |- ( X e. V -> X e. { X } )
21anim1i 592 . . 3 |- ( ( X e. V /\ Y e. A ) -> ( X e. { X } /\ Y e. A ) )
3 eqid 2622 . . 3 |- Y = Y
42, 3jctil 560 . 2 |- ( ( X e. V /\ Y e. A ) -> ( Y = Y /\ ( X e. { X } /\ Y e. A ) ) )
5 2ndconst 7266 . . . . . 6 |- ( X e. V -> ( 2nd |` ( { X } X. A ) ) : ( { X } X. A ) -1-1-onto-> A )
65adantr 481 . . . . 5 |- ( ( X e. V /\ Y e. A ) -> ( 2nd |` ( { X } X. A ) ) : ( { X } X. A ) -1-1-onto-> A )
7 f1ocnv 6149 . . . . 5 |- ( ( 2nd |` ( { X } X. A ) ) : ( { X } X. A ) -1-1-onto-> A -> `' ( 2nd |` ( { X } X. A ) ) : A -1-1-onto-> ( { X } X. A ) )
8 f1ofn 6138 . . . . 5 |- ( `' ( 2nd |` ( { X } X. A ) ) : A -1-1-onto-> ( { X } X. A ) -> `' ( 2nd |` ( { X } X. A ) ) Fn A )
96, 7, 83syl 18 . . . 4 |- ( ( X e. V /\ Y e. A ) -> `' ( 2nd |` ( { X } X. A ) ) Fn A )
10 fnbrfvb 6236 . . . 4 |- ( ( `' ( 2nd |` ( { X } X. A ) ) Fn A /\ Y e. A ) -> ( ( `' ( 2nd |` ( { X } X. A ) ) ` Y ) = <. X , Y >. <-> Y `' ( 2nd |` ( { X } X. A ) ) <. X , Y >. ) )
119, 10sylancom 701 . . 3 |- ( ( X e. V /\ Y e. A ) -> ( ( `' ( 2nd |` ( { X } X. A ) ) ` Y ) = <. X , Y >. <-> Y `' ( 2nd |` ( { X } X. A ) ) <. X , Y >. ) )
12 opex 4932 . . . . . 6 |- <. X , Y >. e. _V
13 brcnvg 5303 . . . . . 6 |- ( ( Y e. A /\ <. X , Y >. e. _V ) -> ( Y `' ( 2nd |` ( { X } X. A ) ) <. X , Y >. <-> <. X , Y >. ( 2nd |` ( { X } X. A ) ) Y ) )
1412, 13mpan2 707 . . . . 5 |- ( Y e. A -> ( Y `' ( 2nd |` ( { X } X. A ) ) <. X , Y >. <-> <. X , Y >. ( 2nd |` ( { X } X. A ) ) Y ) )
1514adantl 482 . . . 4 |- ( ( X e. V /\ Y e. A ) -> ( Y `' ( 2nd |` ( { X } X. A ) ) <. X , Y >. <-> <. X , Y >. ( 2nd |` ( { X } X. A ) ) Y ) )
16 brresg 5405 . . . . . 6 |- ( Y e. A -> ( <. X , Y >. ( 2nd |` ( { X } X. A ) ) Y <-> ( <. X , Y >. 2nd Y /\ <. X , Y >. e. ( { X } X. A ) ) ) )
1716adantl 482 . . . . 5 |- ( ( X e. V /\ Y e. A ) -> ( <. X , Y >. ( 2nd |` ( { X } X. A ) ) Y <-> ( <. X , Y >. 2nd Y /\ <. X , Y >. e. ( { X } X. A ) ) ) )
18 opelxp 5146 . . . . . . 7 |- ( <. X , Y >. e. ( { X } X. A ) <-> ( X e. { X } /\ Y e. A ) )
1918anbi2i 730 . . . . . 6 |- ( ( <. X , Y >. 2nd Y /\ <. X , Y >. e. ( { X } X. A ) ) <-> ( <. X , Y >. 2nd Y /\ ( X e. { X } /\ Y e. A ) ) )
20 br2ndeqg 31673 . . . . . . 7 |- ( ( X e. V /\ Y e. A ) -> ( <. X , Y >. 2nd Y <-> Y = Y ) )
2120anbi1d 741 . . . . . 6 |- ( ( X e. V /\ Y e. A ) -> ( ( <. X , Y >. 2nd Y /\ ( X e. { X } /\ Y e. A ) ) <-> ( Y = Y /\ ( X e. { X } /\ Y e. A ) ) ) )
2219, 21syl5bb 272 . . . . 5 |- ( ( X e. V /\ Y e. A ) -> ( ( <. X , Y >. 2nd Y /\ <. X , Y >. e. ( { X } X. A ) ) <-> ( Y = Y /\ ( X e. { X } /\ Y e. A ) ) ) )
2317, 22bitrd 268 . . . 4 |- ( ( X e. V /\ Y e. A ) -> ( <. X , Y >. ( 2nd |` ( { X } X. A ) ) Y <-> ( Y = Y /\ ( X e. { X } /\ Y e. A ) ) ) )
2415, 23bitrd 268 . . 3 |- ( ( X e. V /\ Y e. A ) -> ( Y `' ( 2nd |` ( { X } X. A ) ) <. X , Y >. <-> ( Y = Y /\ ( X e. { X } /\ Y e. A ) ) ) )
2511, 24bitrd 268 . 2 |- ( ( X e. V /\ Y e. A ) -> ( ( `' ( 2nd |` ( { X } X. A ) ) ` Y ) = <. X , Y >. <-> ( Y = Y /\ ( X e. { X } /\ Y e. A ) ) ) )
264, 25mpbird 247 1 |- ( ( X e. V /\ Y e. A ) -> ( `' ( 2nd |` ( { X } X. A ) ) ` Y ) = <. X , Y >. )
Colors of variables: wff setvar class
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   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-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)
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