Theorem elcnvintab 37908

Description: Two ways of saying a set is an element of the converse of the intersection of a class. (Contributed by RP, 19-Aug-2020.)

Assertion

Ref	Expression
elcnvintab	|- ( A e. `' |^| { x | ph } <-> ( A e. ( _V X. _V ) /\ A. x ( ph -> A e. `' x ) ) )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem elcnvintab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( y e. ( _V X. _V ) |-> <. ( 2nd ` y ) , ( 1st ` y ) >. ) = ( y e. ( _V X. _V ) |-> <. ( 2nd ` y ) , ( 1st ` y ) >. )
2	1	elcnvlem 37907	. 2 |- ( A e. `' |^| { x | ph } <-> ( A e. ( _V X. _V ) /\ ( ( y e. ( _V X. _V ) |-> <. ( 2nd ` y ) , ( 1st ` y ) >. ) ` A ) e. |^| { x | ph } ) )
3	1	elcnvlem 37907	. 2 |- ( A e. `' x <-> ( A e. ( _V X. _V ) /\ ( ( y e. ( _V X. _V ) |-> <. ( 2nd ` y ) , ( 1st ` y ) >. ) ` A ) e. x ) )
4	2, 3	elmapintab 37902	1 |- ( A e. `' |^| { x | ph } <-> ( A e. ( _V X. _V ) /\ A. x ( ph -> A e. `' x ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   {cab 2608   _Vcvv 3200   <.cop 4183   |^|cint 4475   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ` 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-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-int 4476  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-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  cnvintabd  37909