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Theorem imasex 13170
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 2783 . . . 4 |- ( F e. V -> F e. _V )
21adantr 276 . . 3 |- ( ( F e. V /\ R e. W ) -> F e. _V )
3 elex 2783 . . . 4 |- ( R e. W -> R e. _V )
43adantl 277 . . 3 |- ( ( F e. V /\ R e. W ) -> R e. _V )
5 basfn 12923 . . . . . 6 |- Base Fn _V
6 funfvex 5595 . . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
76funfni 5377 . . . . . 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 12921 . . . . . . 7 |- ( Base ` ndx ) e. NN
11 rnexg 4944 . . . . . . . 8 |- ( F e. V -> ran F e. _V )
1211adantr 276 . . . . . . 7 |- ( ( F e. V /\ R e. W ) -> ran F e. _V )
13 opexg 4273 . . . . . . 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 12976 . . . . . . 7 |- ( +g ` ndx ) e. NN
16 vex 2775 . . . . . . . 8 |- v e. _V
17 vex 2775 . . . . . . . . . . . . . . . 16 |- p e. _V
1817a1i 9 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> p e. _V )
19 fvexg 5597 . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . 16 |- q e. _V
2221a1i 9 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> q e. _V )
23 fvexg 5597 . . . . . . . . . . . . . . 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 4273 . . . . . . . . . . . . . 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 12977 . . . . . . . . . . . . . . . . 17 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
2827slotex 12892 . . . . . . . . . . . . . . . 16 |- ( R e. W -> ( +g ` R ) e. _V )
2928adantl 277 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> ( +g ` R ) e. _V )
30 ovexg 5980 . . . . . . . . . . . . . . 15 |- ( ( p e. _V /\ ( +g ` R ) e. _V /\ q e. _V ) -> ( p ( +g ` R ) q ) e. _V )
3118, 29, 22, 30syl3anc 1250 . . . . . . . . . . . . . 14 |- ( ( F e. V /\ R e. W ) -> ( p ( +g ` R ) q ) e. _V )
32 fvexg 5597 . . . . . . . . . . . . . 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 4273 . . . . . . . . . . . . 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 4229 . . . . . . . . . . . 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 2580 . . . . . . . . . 10 |- ( ( F e. V /\ R e. W ) -> A. q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
39 iunexg 6206 . . . . . . . . . 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 2580 . . . . . . . 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 6206 . . . . . . . 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 4273 . . . . . . 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 12997 . . . . . . . 8 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
4746simpri 113 . . . . . . 7 |- ( .r ` ndx ) e. NN
4846slotex 12892 . . . . . . . . . . . . . . . 16 |- ( R e. W -> ( .r ` R ) e. _V )
4948adantl 277 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> ( .r ` R ) e. _V )
50 ovexg 5980 . . . . . . . . . . . . . . 15 |- ( ( p e. _V /\ ( .r ` R ) e. _V /\ q e. _V ) -> ( p ( .r ` R ) q ) e. _V )
5118, 49, 22, 50syl3anc 1250 . . . . . . . . . . . . . 14 |- ( ( F e. V /\ R e. W ) -> ( p ( .r ` R ) q ) e. _V )
52 fvexg 5597 . . . . . . . . . . . . . 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 4273 . . . . . . . . . . . . 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 4229 . . . . . . . . . . . 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 2580 . . . . . . . . . 10 |- ( ( F e. V /\ R e. W ) -> A. q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
59 iunexg 6206 . . . . . . . . . 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 2580 . . . . . . . 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 6206 . . . . . . . 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 4273 . . . . . . 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 4492 . . . . . 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 1250 . . . . 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 1897 . . . 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 4173 . . . 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 4906 . . . . . . 7 |- ( f = F -> ran f = ran F )
7271opeq2d 3826 . . . . . 6 |- ( f = F -> <. ( Base ` ndx ) , ran f >. = <. ( Base ` ndx ) , ran F >. )
73 fveq1 5577 . . . . . . . . . . . 12 |- ( f = F -> ( f ` p ) = ( F ` p ) )
74 fveq1 5577 . . . . . . . . . . . 12 |- ( f = F -> ( f ` q ) = ( F ` q ) )
7573, 74opeq12d 3827 . . . . . . . . . . 11 |- ( f = F -> <. ( f ` p ) , ( f ` q ) >. = <. ( F ` p ) , ( F ` q ) >. )
76 fveq1 5577 . . . . . . . . . . 11 |- ( f = F -> ( f ` ( p ( +g ` r ) q ) ) = ( F ` ( p ( +g ` r ) q ) ) )
7775, 76opeq12d 3827 . . . . . . . . . 10 |- ( f = F -> <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. )
7877sneqd 3646 . . . . . . . . 9 |- ( f = F -> { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } )
7978iuneq2d 3952 . . . . . . . 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 3952 . . . . . . 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 3826 . . . . . 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 5577 . . . . . . . . . . 11 |- ( f = F -> ( f ` ( p ( .r ` r ) q ) ) = ( F ` ( p ( .r ` r ) q ) ) )
8375, 82opeq12d 3827 . . . . . . . . . 10 |- ( f = F -> <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. )
8483sneqd 3646 . . . . . . . . 9 |- ( f = F -> { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } )
8584iuneq2d 3952 . . . . . . . 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 3952 . . . . . . 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 3826 . . . . . 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 3725 . . . . 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 3119 . . . 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 5578 . . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
9190csbeq1d 3100 . . . . 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 2206 . . . . . . 7 |- ( r = R -> <. ( Base ` ndx ) , ran F >. = <. ( Base ` ndx ) , ran F >. )
93 fveq2 5578 . . . . . . . . . . . . . 14 |- ( r = R -> ( +g ` r ) = ( +g ` R ) )
9493oveqd 5963 . . . . . . . . . . . . 13 |- ( r = R -> ( p ( +g ` r ) q ) = ( p ( +g ` R ) q ) )
9594fveq2d 5582 . . . . . . . . . . . 12 |- ( r = R -> ( F ` ( p ( +g ` r ) q ) ) = ( F ` ( p ( +g ` R ) q ) ) )
9695opeq2d 3826 . . . . . . . . . . 11 |- ( r = R -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. )
9796sneqd 3646 . . . . . . . . . 10 |- ( r = R -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } )
9897iuneq2d 3952 . . . . . . . . 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 3952 . . . . . . . 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 3826 . . . . . . 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 5578 . . . . . . . . . . . . . 14 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
102101oveqd 5963 . . . . . . . . . . . . 13 |- ( r = R -> ( p ( .r ` r ) q ) = ( p ( .r ` R ) q ) )
103102fveq2d 5582 . . . . . . . . . . . 12 |- ( r = R -> ( F ` ( p ( .r ` r ) q ) ) = ( F ` ( p ( .r ` R ) q ) ) )
104103opeq2d 3826 . . . . . . . . . . 11 |- ( r = R -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. )
105104sneqd 3646 . . . . . . . . . 10 |- ( r = R -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } )
106105iuneq2d 3952 . . . . . . . . 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 3952 . . . . . . . 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 3826 . . . . . . 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 3725 . . . . . 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 3119 . . . . 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 2238 . . . 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 13167 . . . 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 6082 . . 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 1250 . 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 2282 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 1371   = wceq 1373   e. wcel 2176   A.wral 2484   _Vcvv 2772   [_csb 3093   {csn 3633   {ctp 3635   <.cop 3636   U_ciun 3927   ran crn 4677   Fn wfn 5267   ` cfv 5272  (class class class)co 5946   NNcn 9038   ndxcnx 12862  Slot cslot 12864   Basecbs 12865   +g cplusg 12942   .rcmulr 12943   "s cimas 13164
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1re 8021  ax-addrcl 8024
This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-tp 3641  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-inn 9039  df-2 9097  df-3 9098  df-ndx 12868  df-slot 12869  df-base 12871  df-plusg 12955  df-mulr 12956  df-iimas 13167
This theorem is referenced by:  imasmulr  13174  qusval  13188  qusex  13190  xpsval  13217
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