Theorem mpt2xopoveqd 7347

Description: Value of an operation given by a maps-to rule, where the first argument is a pair and the base set of the second argument is the first component of the first argument, deduction version. (Contributed by Alexander van der Vekens, 11-Oct-2017.)

Hypotheses

Ref	Expression
mpt2xopoveq.f	|- F = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | ph } )
mpt2xopoveqd.1	|- ( ps -> ( V e. X /\ W e. Y ) )
mpt2xopoveqd.2	|- ( ( ps /\ -. K e. V ) -> { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } = (/) )

Assertion

Ref	Expression
mpt2xopoveqd	|- ( ps -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )

Distinct variable groups:   n, K, x, y   n, V, x, y   n, W, x, y   n, X, x, y   n, Y, x, y   x, F

Allowed substitution hints:   ph( x, y, n)   ps( x, y, n)   F( y, n)

Proof of Theorem mpt2xopoveqd

Step	Hyp	Ref	Expression
1		mpt2xopoveq.f	. . . . 5 |- F = ( x e. _V , y e. ( 1st ` x ) |-> { n e. ( 1st ` x ) | ph } )
2	1	mpt2xopoveq 7345	. . . 4 |- ( ( ( V e. X /\ W e. Y ) /\ K e. V ) -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )
3	2	ex 450	. . 3 |- ( ( V e. X /\ W e. Y ) -> ( K e. V -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } ) )
4		mpt2xopoveqd.1	. . 3 |- ( ps -> ( V e. X /\ W e. Y ) )
5	3, 4	syl11 33	. 2 |- ( K e. V -> ( ps -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } ) )
6		df-nel 2898	. . . . . 6 |- ( K e/ V <-> -. K e. V )
7	1	mpt2xopynvov0 7344	. . . . . 6 |- ( K e/ V -> ( <. V , W >. F K ) = (/) )
8	6, 7	sylbir 225	. . . . 5 |- ( -. K e. V -> ( <. V , W >. F K ) = (/) )
9	8	adantr 481	. . . 4 |- ( ( -. K e. V /\ ps ) -> ( <. V , W >. F K ) = (/) )
10		mpt2xopoveqd.2	. . . . . 6 |- ( ( ps /\ -. K e. V ) -> { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } = (/) )
11	10	eqcomd 2628	. . . . 5 |- ( ( ps /\ -. K e. V ) -> (/) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )
12	11	ancoms 469	. . . 4 |- ( ( -. K e. V /\ ps ) -> (/) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )
13	9, 12	eqtrd 2656	. . 3 |- ( ( -. K e. V /\ ps ) -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )
14	13	ex 450	. 2 |- ( -. K e. V -> ( ps -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } ) )
15	5, 14	pm2.61i 176	1 |- ( ps -> ( <. V , W >. F K ) = { n e. V | [. <. V , W >. / x ]. [. K / y ]. ph } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   e/ wnel 2897   {crab 2916   _Vcvv 3200   [.wsbc 3435   (/)c0 3915   <.cop 4183   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   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-nel 2898  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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by: (None)