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Theorem imasex 13603
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 2833 . . . 4 |- ( F e. V -> F e. _V )
21adantr 276 . . 3 |- ( ( F e. V /\ R e. W ) -> F e. _V )
3 elex 2833 . . . 4 |- ( R e. W -> R e. _V )
43adantl 277 . . 3 |- ( ( F e. V /\ R e. W ) -> R e. _V )
5 basfn 13389 . . . . . 6 |- Base Fn _V
6 funfvex 5707 . . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
76funfni 5478 . . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
85, 3, 7sylancr 418 . . . . 5 |- ( R e. W -> ( Base ` R ) e. _V )
98adantl 277 . . . 4 |- ( ( F e. V /\ R e. W ) -> ( Base ` R ) e. _V )
10 basendxnn 13386 . . . . . . 7 |- ( Base ` ndx ) e. NN
11 rnexg 5042 . . . . . . . 8 |- ( F e. V -> ran F e. _V )
1211adantr 276 . . . . . . 7 |- ( ( F e. V /\ R e. W ) -> ran F e. _V )
13 opexg 4363 . . . . . . 7 |- ( ( ( Base ` ndx ) e. NN /\ ran F e. _V ) -> <. ( Base ` ndx ) , ran F >. e. _V )
1410, 12, 13sylancr 418 . . . . . 6 |- ( ( F e. V /\ R e. W ) -> <. ( Base ` ndx ) , ran F >. e. _V )
15 plusgndxnn 13442 . . . . . . 7 |- ( +g ` ndx ) e. NN
16 vex 2824 . . . . . . . 8 |- v e. _V
17 vex 2824 . . . . . . . . . . . . . . . 16 |- p e. _V
1817a1i 9 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> p e. _V )
19 fvexg 5709 . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . 16 |- q e. _V
2221a1i 9 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> q e. _V )
23 fvexg 5709 . . . . . . . . . . . . . . 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 4363 . . . . . . . . . . . . . 14 |- ( ( ( F ` p ) e. _V /\ ( F ` q ) e. _V ) -> <. ( F ` p ) , ( F ` q ) >. e. _V )
2620, 24, 25syl2anc 415 . . . . . . . . . . . . 13 |- ( ( F e. V /\ R e. W ) -> <. ( F ` p ) , ( F ` q ) >. e. _V )
27 plusgslid 13443 . . . . . . . . . . . . . . . . 17 |- ( +g = Slot ( +g ` ndx ) /\ ( +g ` ndx ) e. NN )
2827slotex 13357 . . . . . . . . . . . . . . . 16 |- ( R e. W -> ( +g ` R ) e. _V )
2928adantl 277 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> ( +g ` R ) e. _V )
30 ovexg 6109 . . . . . . . . . . . . . . 15 |- ( ( p e. _V /\ ( +g ` R ) e. _V /\ q e. _V ) -> ( p ( +g ` R ) q ) e. _V )
3118, 29, 22, 30syl3anc 1278 . . . . . . . . . . . . . 14 |- ( ( F e. V /\ R e. W ) -> ( p ( +g ` R ) q ) e. _V )
32 fvexg 5709 . . . . . . . . . . . . . 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 4363 . . . . . . . . . . . . 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 415 . . . . . . . . . . . 12 |- ( ( F e. V /\ R e. W ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. e. _V )
36 snexg 4316 . . . . . . . . . . . 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 2624 . . . . . . . . . 10 |- ( ( F e. V /\ R e. W ) -> A. q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
39 iunexg 6338 . . . . . . . . . 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 418 . . . . . . . . 9 |- ( ( F e. V /\ R e. W ) -> U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } e. _V )
4140ralrimivw 2624 . . . . . . . 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 6338 . . . . . . . 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 418 . . . . . . 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 4363 . . . . . . 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 418 . . . . . 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 13463 . . . . . . . 8 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
4746simpri 113 . . . . . . 7 |- ( .r ` ndx ) e. NN
4846slotex 13357 . . . . . . . . . . . . . . . 16 |- ( R e. W -> ( .r ` R ) e. _V )
4948adantl 277 . . . . . . . . . . . . . . 15 |- ( ( F e. V /\ R e. W ) -> ( .r ` R ) e. _V )
50 ovexg 6109 . . . . . . . . . . . . . . 15 |- ( ( p e. _V /\ ( .r ` R ) e. _V /\ q e. _V ) -> ( p ( .r ` R ) q ) e. _V )
5118, 49, 22, 50syl3anc 1278 . . . . . . . . . . . . . 14 |- ( ( F e. V /\ R e. W ) -> ( p ( .r ` R ) q ) e. _V )
52 fvexg 5709 . . . . . . . . . . . . . 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 4363 . . . . . . . . . . . . 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 415 . . . . . . . . . . . 12 |- ( ( F e. V /\ R e. W ) -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. e. _V )
56 snexg 4316 . . . . . . . . . . . 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 2624 . . . . . . . . . 10 |- ( ( F e. V /\ R e. W ) -> A. q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
59 iunexg 6338 . . . . . . . . . 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 418 . . . . . . . . 9 |- ( ( F e. V /\ R e. W ) -> U_ q e. v { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } e. _V )
6160ralrimivw 2624 . . . . . . . 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 6338 . . . . . . . 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 418 . . . . . . 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 4363 . . . . . . 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 418 . . . . . 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 4585 . . . . . 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 1278 . . . . 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 1927 . . . 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 4256 . . . 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 415 . . 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 5004 . . . . . . 7 |- ( f = F -> ran f = ran F )
7271opeq2d 3906 . . . . . 6 |- ( f = F -> <. ( Base ` ndx ) , ran f >. = <. ( Base ` ndx ) , ran F >. )
73 fveq1 5689 . . . . . . . . . . . 12 |- ( f = F -> ( f ` p ) = ( F ` p ) )
74 fveq1 5689 . . . . . . . . . . . 12 |- ( f = F -> ( f ` q ) = ( F ` q ) )
7573, 74opeq12d 3907 . . . . . . . . . . 11 |- ( f = F -> <. ( f ` p ) , ( f ` q ) >. = <. ( F ` p ) , ( F ` q ) >. )
76 fveq1 5689 . . . . . . . . . . 11 |- ( f = F -> ( f ` ( p ( +g ` r ) q ) ) = ( F ` ( p ( +g ` r ) q ) ) )
7775, 76opeq12d 3907 . . . . . . . . . 10 |- ( f = F -> <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. )
7877sneqd 3718 . . . . . . . . 9 |- ( f = F -> { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } )
7978iuneq2d 4032 . . . . . . . 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 4032 . . . . . . 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 3906 . . . . . 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 5689 . . . . . . . . . . 11 |- ( f = F -> ( f ` ( p ( .r ` r ) q ) ) = ( F ` ( p ( .r ` r ) q ) ) )
8375, 82opeq12d 3907 . . . . . . . . . 10 |- ( f = F -> <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. )
8483sneqd 3718 . . . . . . . . 9 |- ( f = F -> { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } )
8584iuneq2d 4032 . . . . . . . 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 4032 . . . . . . 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 3906 . . . . . 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 3799 . . . . 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 3173 . . . 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 5690 . . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
9190csbeq1d 3154 . . . . 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 2239 . . . . . . 7 |- ( r = R -> <. ( Base ` ndx ) , ran F >. = <. ( Base ` ndx ) , ran F >. )
93 fveq2 5690 . . . . . . . . . . . . . 14 |- ( r = R -> ( +g ` r ) = ( +g ` R ) )
9493oveqd 6092 . . . . . . . . . . . . 13 |- ( r = R -> ( p ( +g ` r ) q ) = ( p ( +g ` R ) q ) )
9594fveq2d 5694 . . . . . . . . . . . 12 |- ( r = R -> ( F ` ( p ( +g ` r ) q ) ) = ( F ` ( p ( +g ` R ) q ) ) )
9695opeq2d 3906 . . . . . . . . . . 11 |- ( r = R -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. )
9796sneqd 3718 . . . . . . . . . 10 |- ( r = R -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( +g ` R ) q ) ) >. } )
9897iuneq2d 4032 . . . . . . . . 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 4032 . . . . . . . 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 3906 . . . . . . 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 5690 . . . . . . . . . . . . . 14 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
102101oveqd 6092 . . . . . . . . . . . . 13 |- ( r = R -> ( p ( .r ` r ) q ) = ( p ( .r ` R ) q ) )
103102fveq2d 5694 . . . . . . . . . . . 12 |- ( r = R -> ( F ` ( p ( .r ` r ) q ) ) = ( F ` ( p ( .r ` R ) q ) ) )
104103opeq2d 3906 . . . . . . . . . . 11 |- ( r = R -> <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. )
105104sneqd 3718 . . . . . . . . . 10 |- ( r = R -> { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p ( .r ` R ) q ) ) >. } )
106105iuneq2d 4032 . . . . . . . . 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 4032 . . . . . . . 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 3906 . . . . . . 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 3799 . . . . . 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 3173 . . . . 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 2271 . . . 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 13601 . . . 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 6213 . . 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 1278 . 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 2315 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 1400   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   [_csb 3147   {csn 3705   {ctp 3707   <.cop 3708   U_ciun 4007   ran crn 4770   Fn wfn 5367   ` cfv 5372  (class class class)co 6075   NNcn 9283   ndxcnx 13327  Slot cslot 13329   Basecbs 13330   +g cplusg 13408   .rcmulr 13409   "s cimas 13599
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266
This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-tp 3713  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-inn 9284  df-2 9342  df-3 9343  df-ndx 13333  df-slot 13334  df-base 13336  df-plusg 13421  df-mulr 13422  df-iimas 13601
This theorem is referenced by:  imasmulr  13607  qusval  13621  qusex  13623  xpsval  14178
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