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Theorem imasex 13007
Description: Existence of the image structure. (Contributed by Jim Kingdon, 13-Mar-2025.)
Assertion
Ref Expression
imasex |- ( ( F e. V /\ R e. W ) -> ( F "s R ) e. _V )
Proof of Theorem imasex
Dummy variables f p q r v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2774 . . . 4 |- ( F e. V -> F e. _V )
21adantr 276 . . 3 |- ( ( F e. V /\ R e. W ) -> F e. _V )
3 elex 2774 . . . 4 |- ( R e. W -> R e. _V )
43adantl 277 . . 3 |- ( ( F e. V /\ R e. W ) -> R e. _V )
5 basfn 12761 . . . . . 6 |- Base Fn _V
6 funfvex 5578 . . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
76funfni 5361 . . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
85, 3, 7sylancr 414 . . . . 5 |- ( R e. W -> ( Base ` R ) e. _V )
98adantl 277 . . . 4 |- ( ( F e. V /\ R e. W ) -> ( Base ` R ) e. _V )
10 basendxnn 12759 . . . . . . 7 |- ( Base ` ndx ) e. NN
11 rnexg 4932 . . . . . . . 8 |- ( F e. V -> ran F e. _V )
1211adantr 276 . . . . . . 7 |- ( ( F e. V /\ R e. W ) -> ran F e. _V )
13 opexg 4262 . . . . . . 7 |- ( ( ( Base ` ndx ) e. NN /\ ran F e. _V ) -> <. ( Base ` ndx ) , ran F >. e. _V )
1410, 12, 13sylancr 414 . . . . . 6 |- ( ( F e. V /\ R e. W ) -> <. ( Base ` ndx ) , ran F >. e. _V )
15 plusgndxnn 12814 . . . . . . 7 |- ( +g ` ndx ) e. NN
16 vex 2766 . . . . . . . 8 |- v e. _V
17 vex 2766 . . . . . . . . . . . . . . . 16 |- p e. _V
1817a1i 9 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> p e. _V )
19 fvexg 5580 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ p e. _V ) -> ( F ` p ) e. _V )
2018, 19syldan 282 . . . . . . . . . . . . . 14 |- ( ( F e. V /\ R e. W ) -> ( F ` p ) e. _V )
21 vex 2766 . . . . . . . . . . . . . . . 16 |- q e. _V
2221a1i 9 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> q e. _V )
23 fvexg 5580 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ q e. _V ) -> ( F ` q ) e. _V )
2422, 23syldan 282 . . . . . . . . . . . . . 14 |- ( ( F e. V /\ R e. W ) -> ( F ` q ) e. _V )
25 opexg 4262 . . . . . . . . . . . . . 14 |- ( ( ( F ` p ) e. _V /\ ( F ` q ) e. _V ) -> <. ( F ` p ) , ( F ` q ) >. e. _V )
2620, 24, 25syl2anc 411 . . . . . . . . . . . . 13 |- ( ( F e. V /\ R e. W ) -> <. ( F ` p ) , ( F ` q ) >. e. _V )
27 plusgslid 12815 . . . . . . . . . . . . . . . . 17 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
2827slotex 12730 . . . . . . . . . . . . . . . 16 |- ( R e. W -> ( +g ` R ) e. _V )
2928adantl 277 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> ( +g ` R ) e. _V )
30 ovexg 5959 . . . . . . . . . . . . . . 15 |- ( ( p e. _V /\ ( +g ` R ) e. _V /\ q e. _V ) -> ( p ( +g ` R ) q ) e. _V )
3118, 29, 22, 30syl3anc 1249 . . . . . . . . . . . . . 14 |- ( ( F e. V /\ R e. W ) -> ( p ( +g ` R ) q ) e. _V )
32 fvexg 5580 . . . . . . . . . . . . . 14 |- ( ( F e. V /\ ( p ( +g ` R ) q ) e. _V ) -> ( F ` ( p ( +g ` R ) q ) ) e. _V )
3331, 32syldan 282 . . . . . . . . . . . . 13 |- ( ( F e. V /\ R e. W ) -> ( F ` ( p ( +g ` R ) q ) ) e. _V )
34 opexg 4262 . . . . . . . . . . . . 13 |- ( ( <. ( F ` p ) , ( F ` q ) >. e. _V /\ ( F ` ( p ( +g ` R ) q ) ) e. _V ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. e. _V )
3526, 33, 34syl2anc 411 . . . . . . . . . . . 12 |- ( ( F e. V /\ R e. W ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. e. _V )
36 snexg 4218 . . . . . . . . . . . 12 |- ( <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. e. _V -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
3735, 36syl 14 . . . . . . . . . . 11 |- ( ( F e. V /\ R e. W ) -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
3837ralrimivw 2571 . . . . . . . . . 10 |- ( ( F e. V /\ R e. W ) -> A. q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
39 iunexg 6185 . . . . . . . . . 10 |- ( ( v e. _V /\ A. q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V ) -> U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
4016, 38, 39sylancr 414 . . . . . . . . 9 |- ( ( F e. V /\ R e. W ) -> U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
4140ralrimivw 2571 . . . . . . . 8 |- ( ( F e. V /\ R e. W ) -> A. p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
42 iunexg 6185 . . . . . . . 8 |- ( ( v e. _V /\ A. p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V ) -> U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
4316, 41, 42sylancr 414 . . . . . . 7 |- ( ( F e. V /\ R e. W ) -> U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
44 opexg 4262 . . . . . . 7 |- ( ( ( +g ` ndx ) e. NN /\ U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V ) -> <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. e. _V )
4515, 43, 44sylancr 414 . . . . . 6 |- ( ( F e. V /\ R e. W ) -> <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. e. _V )
46 mulrslid 12834 . . . . . . . 8 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
4746simpri 113 . . . . . . 7 |- ( .r ` ndx ) e. NN
4846slotex 12730 . . . . . . . . . . . . . . . 16 |- ( R e. W -> ( .r ` R ) e. _V )
4948adantl 277 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> ( .r ` R ) e. _V )
50 ovexg 5959 . . . . . . . . . . . . . . 15 |- ( ( p e. _V /\ ( .r ` R ) e. _V /\ q e. _V ) -> ( p ( .r ` R ) q ) e. _V )
5118, 49, 22, 50syl3anc 1249 . . . . . . . . . . . . . 14 |- ( ( F e. V /\ R e. W ) -> ( p ( .r ` R ) q ) e. _V )
52 fvexg 5580 . . . . . . . . . . . . . 14 |- ( ( F e. V /\ ( p ( .r ` R ) q ) e. _V ) -> ( F ` ( p ( .r ` R ) q ) ) e. _V )
5351, 52syldan 282 . . . . . . . . . . . . 13 |- ( ( F e. V /\ R e. W ) -> ( F ` ( p ( .r ` R ) q ) ) e. _V )
54 opexg 4262 . . . . . . . . . . . . 13 |- ( ( <. ( F ` p ) , ( F ` q ) >. e. _V /\ ( F ` ( p ( .r ` R ) q ) ) e. _V ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. e. _V )
5526, 53, 54syl2anc 411 . . . . . . . . . . . 12 |- ( ( F e. V /\ R e. W ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. e. _V )
56 snexg 4218 . . . . . . . . . . . 12 |- ( <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. e. _V -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
5755, 56syl 14 . . . . . . . . . . 11 |- ( ( F e. V /\ R e. W ) -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
5857ralrimivw 2571 . . . . . . . . . 10 |- ( ( F e. V /\ R e. W ) -> A. q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
59 iunexg 6185 . . . . . . . . . 10 |- ( ( v e. _V /\ A. q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V ) -> U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
6016, 58, 59sylancr 414 . . . . . . . . 9 |- ( ( F e. V /\ R e. W ) -> U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
6160ralrimivw 2571 . . . . . . . 8 |- ( ( F e. V /\ R e. W ) -> A. p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
62 iunexg 6185 . . . . . . . 8 |- ( ( v e. _V /\ A. p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V ) -> U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
6316, 61, 62sylancr 414 . . . . . . 7 |- ( ( F e. V /\ R e. W ) -> U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
64 opexg 4262 . . . . . . 7 |- ( ( ( .r ` ndx ) e. NN /\ U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V ) -> <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. e. _V )
6547, 63, 64sylancr 414 . . . . . 6 |- ( ( F e. V /\ R e. W ) -> <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. e. _V )
66 tpexg 4480 . . . . . 6 |- ( ( <. ( Base ` ndx ) , ran F >. e. _V /\ <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. e. _V /\ <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. e. _V ) -> { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } e. _V )
6714, 45, 65, 66syl3anc 1249 . . . . 5 |- ( ( F e. V /\ R e. W ) -> { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } e. _V )
6867alrimiv 1888 . . . 4 |- ( ( F e. V /\ R e. W ) -> A. v { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } e. _V )
69 csbexga 4162 . . . 4 |- ( ( ( Base ` R ) e. _V /\ A. v { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } e. _V ) -> [_ ( Base ` R ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } e. _V )
709, 68, 69syl2anc 411 . . 3 |- ( ( F e. V /\ R e. W ) -> [_ ( Base ` R ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } e. _V )
71 rneq 4894 . . . . . . 7 |- ( f = F -> ran f = ran F )
7271opeq2d 3816 . . . . . 6 |- ( f = F -> <. ( Base ` ndx ) , ran f >. = <. ( Base ` ndx ) , ran F >. )
73 fveq1 5560 . . . . . . . . . . . 12 |- ( f = F -> ( f ` p ) = ( F ` p ) )
74 fveq1 5560 . . . . . . . . . . . 12 |- ( f = F -> ( f ` q ) = ( F ` q ) )
7573, 74opeq12d 3817 . . . . . . . . . . 11 |- ( f = F -> <. ( f ` p ) , ( f ` q ) >. = <. ( F ` p ) , ( F ` q ) >. )
76 fveq1 5560 . . . . . . . . . . 11 |- ( f = F -> ( f ` ( p ( +g ` r ) q ) ) = ( F ` ( p ( +g ` r ) q ) ) )
7775, 76opeq12d 3817 . . . . . . . . . 10 |- ( f = F -> <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. )
7877sneqd 3636 . . . . . . . . 9 |- ( f = F -> { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } )
7978iuneq2d 3942 . . . . . . . 8 |- ( f = F -> U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } )
8079iuneq2d 3942 . . . . . . 7 |- ( f = F -> U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } )
8180opeq2d 3816 . . . . . 6 |- ( f = F -> <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. = <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } >. )
82 fveq1 5560 . . . . . . . . . . 11 |- ( f = F -> ( f ` ( p ( .r ` r ) q ) ) = ( F ` ( p ( .r ` r ) q ) ) )
8375, 82opeq12d 3817 . . . . . . . . . 10 |- ( f = F -> <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. )
8483sneqd 3636 . . . . . . . . 9 |- ( f = F -> { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } )
8584iuneq2d 3942 . . . . . . . 8 |- ( f = F -> U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } )
8685iuneq2d 3942 . . . . . . 7 |- ( f = F -> U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } )
8786opeq2d 3816 . . . . . 6 |- ( f = F -> <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. = <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } >. )
8872, 81, 87tpeq123d 3715 . . . . 5 |- ( f = F -> { <. ( Base ` ndx ) , ran f >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. } = { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } >. } )
8988csbeq2dv 3110 . . . 4 |- ( f = F -> [_ ( Base ` r ) / v ]_ { <. ( Base ` ndx ) , ran f >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. } = [_ ( Base ` r ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } >. } )
90 fveq2 5561 . . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
9190csbeq1d 3091 . . . . 5 |- ( r = R -> [_ ( Base ` r ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } >. } = [_ ( Base ` R ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } >. } )
92 eqidd 2197 . . . . . . 7 |- ( r = R -> <. ( Base ` ndx ) , ran F >. = <. ( Base ` ndx ) , ran F >. )
93 fveq2 5561 . . . . . . . . . . . . . 14 |- ( r = R -> ( +g ` r ) = ( +g ` R ) )
9493oveqd 5942 . . . . . . . . . . . . 13 |- ( r = R -> ( p ( +g ` r ) q ) = ( p ( +g ` R ) q ) )
9594fveq2d 5565 . . . . . . . . . . . 12 |- ( r = R -> ( F ` ( p ( +g ` r ) q ) ) = ( F ` ( p ( +g ` R ) q ) ) )
9695opeq2d 3816 . . . . . . . . . . 11 |- ( r = R -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. )
9796sneqd 3636 . . . . . . . . . 10 |- ( r = R -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } )
9897iuneq2d 3942 . . . . . . . . 9 |- ( r = R -> U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } = U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } )
9998iuneq2d 3942 . . . . . . . 8 |- ( r = R -> U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } = U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } )
10099opeq2d 3816 . . . . . . 7 |- ( r = R -> <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } >. = <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. )
101 fveq2 5561 . . . . . . . . . . . . . 14 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
102101oveqd 5942 . . . . . . . . . . . . 13 |- ( r = R -> ( p ( .r ` r ) q ) = ( p ( .r ` R ) q ) )
103102fveq2d 5565 . . . . . . . . . . . 12 |- ( r = R -> ( F ` ( p ( .r ` r ) q ) ) = ( F ` ( p ( .r ` R ) q ) ) )
104103opeq2d 3816 . . . . . . . . . . 11 |- ( r = R -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. )
105104sneqd 3636 . . . . . . . . . 10 |- ( r = R -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } )
106105iuneq2d 3942 . . . . . . . . 9 |- ( r = R -> U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } = U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } )
107106iuneq2d 3942 . . . . . . . 8 |- ( r = R -> U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } = U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } )
108107opeq2d 3816 . . . . . . 7 |- ( r = R -> <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } >. = <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. )
10992, 100, 108tpeq123d 3715 . . . . . 6 |- ( r = R -> { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } >. } = { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } )
110109csbeq2dv 3110 . . . . 5 |- ( r = R -> [_ ( Base ` R ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } >. } = [_ ( Base ` R ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } )
11191, 110eqtrd 2229 . . . 4 |- ( r = R -> [_ ( Base ` r ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } >. } = [_ ( Base ` R ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } )
112 df-iimas 13004 . . . 4 |- "s = ( f e. _V , r e. _V |-> [_ ( Base ` r ) / v ]_ { <. ( Base ` ndx ) , ran f >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. } )
11389, 111, 112ovmpog 6061 . . 3 |- ( ( F e. _V /\ R e. _V /\ [_ ( Base ` R ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } e. _V ) -> ( F "s R ) = [_ ( Base ` R ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } )
1142, 4, 70, 113syl3anc 1249 . 2 |- ( ( F e. V /\ R e. W ) -> ( F "s R ) = [_ ( Base ` R ) / v ]_ { <. ( Base ` ndx ) , ran F >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } >. } )
115114, 70eqeltrd 2273 1 |- ( ( F e. V /\ R e. W ) -> ( F "s R ) e. _V )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   [_csb 3084   {csn 3623   {ctp 3625   <.cop 3626   U_ciun 3917   ran crn 4665   Fn wfn 5254   ` cfv 5259  (class class class)co 5925   NNcn 9007   ndxcnx 12700  Slot cslot 12702   Basecbs 12703   +g cplusg 12780   .rcmulr 12781   "s cimas 13001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1re 7990  ax-addrcl 7993
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-tp 3631  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-inn 9008  df-2 9066  df-3 9067  df-ndx 12706  df-slot 12707  df-base 12709  df-plusg 12793  df-mulr 12794  df-iimas 13004
This theorem is referenced by:  imasmulr  13011  qusval  13025  qusex  13027  xpsval  13054
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