Theorem fpar 5085

Description: Merge two functions in parallel. Use as the second argument of a composition with a (2-place) operation to build compound operations such as z = ((sqr` x) + (abs` y)).

Hypothesis

Ref	Expression
fpar.1	|- H = ((`'(1st |` (_V X. _V)) o. (F o. (1st |` (_V X. _V)))) i^i (`'(2nd |` (_V X. _V)) o. (G o. (2nd |` (_V X. _V)))))

Assertion

Ref	Expression
fpar	|- ((F Fn A /\ G Fn B) -> H = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = <.(F` x), (G` y)>.)})

Distinct variable groups:   x,y,z,A   x,B,y,z   x,F,y,z   x,G,y,z

Proof of Theorem fpar

Step	Hyp	Ref	Expression
1		fpar.1	. . 3 |- H = ((`'(1st |` (_V X. _V)) o. (F o. (1st |` (_V X. _V)))) i^i (`'(2nd |` (_V X. _V)) o. (G o. (2nd |` (_V X. _V)))))
2	1	a1i 8	. 2 |- ((F Fn A /\ G Fn B) -> H = ((`'(1st |` (_V X. _V)) o. (F o. (1st |` (_V X. _V)))) i^i (`'(2nd |` (_V X. _V)) o. (G o. (2nd |` (_V X. _V))))))
3		fparlem3 5083	. . 3 |- (F Fn A -> (`'(1st |` (_V X. _V)) o. (F o. (1st |` (_V X. _V)))) = U_x e. A (({x} X. _V) X. ({(F` x)} X. _V)))
4		fparlem4 5084	. . 3 |- (G Fn B -> (`'(2nd |` (_V X. _V)) o. (G o. (2nd |` (_V X. _V)))) = U_y e. B ((_V X. {y}) X. (_V X. {(G` y)})))
5	3, 4	ineqan12d 2799	. 2 |- ((F Fn A /\ G Fn B) -> ((`'(1st |` (_V X. _V)) o. (F o. (1st |` (_V X. _V)))) i^i (`'(2nd |` (_V X. _V)) o. (G o. (2nd |` (_V X. _V))))) = (U_x e. A (({x} X. _V) X. ({(F` x)} X. _V)) i^i U_y e. B ((_V X. {y}) X. (_V X. {(G` y)}))))
6		inxp 4109	. . . . . . . . 9 |- ((({x} X. _V) X. ({(F` x)} X. _V)) i^i ((_V X. {y}) X. (_V X. {(G` y)}))) = ((({x} X. _V) i^i (_V X. {y})) X. (({(F` x)} X. _V) i^i (_V X. {(G` y)})))
7		inxp 4109	. . . . . . . . . . . 12 |- (({x} X. _V) i^i (_V X. {y})) = (({x} i^i _V) X. (_V i^i {y}))
8		inv1 2898	. . . . . . . . . . . . . 14 |- ({x} i^i _V) = {x}
9	8	xpeq1i 4021	. . . . . . . . . . . . 13 |- (({x} i^i _V) X. (_V i^i {y})) = ({x} X. (_V i^i {y}))
10		incom 2787	. . . . . . . . . . . . . . 15 |- (_V i^i {y}) = ({y} i^i _V)
11		inv1 2898	. . . . . . . . . . . . . . 15 |- ({y} i^i _V) = {y}
12	10, 11	eqtri 1908	. . . . . . . . . . . . . 14 |- (_V i^i {y}) = {y}
13	12	xpeq2i 4022	. . . . . . . . . . . . 13 |- ({x} X. (_V i^i {y})) = ({x} X. {y})
14	9, 13	eqtri 1908	. . . . . . . . . . . 12 |- (({x} i^i _V) X. (_V i^i {y})) = ({x} X. {y})
15		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
16		visset 2295	. . . . . . . . . . . . 13 |- y e. _V
17	15, 16	xpsn 4808	. . . . . . . . . . . 12 |- ({x} X. {y}) = {<.x, y>.}
18	7, 14, 17	3eqtri 1912	. . . . . . . . . . 11 |- (({x} X. _V) i^i (_V X. {y})) = {<.x, y>.}
19	18	xpeq1i 4021	. . . . . . . . . 10 |- ((({x} X. _V) i^i (_V X. {y})) X. (({(F` x)} X. _V) i^i (_V X. {(G` y)}))) = ({<.x, y>.} X. (({(F` x)} X. _V) i^i (_V X. {(G` y)})))
20		inxp 4109	. . . . . . . . . . . 12 |- (({(F` x)} X. _V) i^i (_V X. {(G` y)})) = (({(F` x)} i^i _V) X. (_V i^i {(G` y)}))
21		inv1 2898	. . . . . . . . . . . . . 14 |- ({(F` x)} i^i _V) = {(F` x)}
22	21	xpeq1i 4021	. . . . . . . . . . . . 13 |- (({(F` x)} i^i _V) X. (_V i^i {(G` y)})) = ({(F` x)} X. (_V i^i {(G` y)}))
23		incom 2787	. . . . . . . . . . . . . . 15 |- (_V i^i {(G` y)}) = ({(G` y)} i^i _V)
24		inv1 2898	. . . . . . . . . . . . . . 15 |- ({(G` y)} i^i _V) = {(G` y)}
25	23, 24	eqtri 1908	. . . . . . . . . . . . . 14 |- (_V i^i {(G` y)}) = {(G` y)}
26	25	xpeq2i 4022	. . . . . . . . . . . . 13 |- ({(F` x)} X. (_V i^i {(G` y)})) = ({(F` x)} X. {(G` y)})
27	22, 26	eqtri 1908	. . . . . . . . . . . 12 |- (({(F` x)} i^i _V) X. (_V i^i {(G` y)})) = ({(F` x)} X. {(G` y)})
28		fvex 4689	. . . . . . . . . . . . 13 |- (F` x) e. _V
29		fvex 4689	. . . . . . . . . . . . 13 |- (G` y) e. _V
30	28, 29	xpsn 4808	. . . . . . . . . . . 12 |- ({(F` x)} X. {(G` y)}) = {<.(F` x), (G` y)>.}
31	20, 27, 30	3eqtri 1912	. . . . . . . . . . 11 |- (({(F` x)} X. _V) i^i (_V X. {(G` y)})) = {<.(F` x), (G` y)>.}
32	31	xpeq2i 4022	. . . . . . . . . 10 |- ({<.x, y>.} X. (({(F` x)} X. _V) i^i (_V X. {(G` y)}))) = ({<.x, y>.} X. {<.(F` x), (G` y)>.})
33	19, 32	eqtri 1908	. . . . . . . . 9 |- ((({x} X. _V) i^i (_V X. {y})) X. (({(F` x)} X. _V) i^i (_V X. {(G` y)}))) = ({<.x, y>.} X. {<.(F` x), (G` y)>.})
34		opex 3527	. . . . . . . . . 10 |- <.x, y>. e. _V
35		opex 3527	. . . . . . . . . 10 |- <.(F` x), (G` y)>. e. _V
36	34, 35	xpsn 4808	. . . . . . . . 9 |- ({<.x, y>.} X. {<.(F` x), (G` y)>.}) = {<.<.x, y>., <.(F` x), (G` y)>.>.}
37	6, 33, 36	3eqtri 1912	. . . . . . . 8 |- ((({x} X. _V) X. ({(F` x)} X. _V)) i^i ((_V X. {y}) X. (_V X. {(G` y)}))) = {<.<.x, y>., <.(F` x), (G` y)>.>.}
38	37	a1i 8	. . . . . . 7 |- (y e. B -> ((({x} X. _V) X. ({(F` x)} X. _V)) i^i ((_V X. {y}) X. (_V X. {(G` y)}))) = {<.<.x, y>., <.(F` x), (G` y)>.>.})
39	38	iuneq2i 3276	. . . . . 6 |- U_y e. B ((({x} X. _V) X. ({(F` x)} X. _V)) i^i ((_V X. {y}) X. (_V X. {(G` y)}))) = U_y e. B {<.<.x, y>., <.(F` x), (G` y)>.>.}
40	39	a1i 8	. . . . 5 |- (x e. A -> U_y e. B ((({x} X. _V) X. ({(F` x)} X. _V)) i^i ((_V X. {y}) X. (_V X. {(G` y)}))) = U_y e. B {<.<.x, y>., <.(F` x), (G` y)>.>.})
41	40	iuneq2i 3276	. . . 4 |- U_x e. A U_y e. B ((({x} X. _V) X. ({(F` x)} X. _V)) i^i ((_V X. {y}) X. (_V X. {(G` y)}))) = U_x e. A U_y e. B {<.<.x, y>., <.(F` x), (G` y)>.>.}
42		2iunin 3323	. . . 4 |- U_x e. A U_y e. B ((({x} X. _V) X. ({(F` x)} X. _V)) i^i ((_V X. {y}) X. (_V X. {(G` y)}))) = (U_x e. A (({x} X. _V) X. ({(F` x)} X. _V)) i^i U_y e. B ((_V X. {y}) X. (_V X. {(G` y)})))
43	35	iunfoprab 5072	. . . 4 |- U_x e. A U_y e. B {<.<.x, y>., <.(F` x), (G` y)>.>.} = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = <.(F` x), (G` y)>.)}
44	41, 42, 43	3eqtr3i 1918	. . 3 |- (U_x e. A (({x} X. _V) X. ({(F` x)} X. _V)) i^i U_y e. B ((_V X. {y}) X. (_V X. {(G` y)}))) = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = <.(F` x), (G` y)>.)}
45	44	a1i 8	. 2 |- ((F Fn A /\ G Fn B) -> (U_x e. A (({x} X. _V) X. ({(F` x)} X. _V)) i^i U_y e. B ((_V X. {y}) X. (_V X. {(G` y)}))) = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = <.(F` x), (G` y)>.)})
46	2, 5, 45	3eqtrd 1929	1 |- ((F Fn A /\ G Fn B) -> H = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = <.(F` x), (G` y)>.)})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   i^i cin 2592  {csn 3044  <.cop 3046  U_ciun 3255   X. cxp 3984  `'ccnv 3985   |` cres 3988   o. ccom 3990   Fn wfn 3993  ` cfv 3998  {copab2 4885  1stc1st 5018  2ndc2nd 5019

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-oprab 4887  df-1st 5020  df-2nd 5021

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