Theorem opeliunxp 4338

Description: Membership in a union of cross products. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 1-Jan-2017.)

Assertion

Ref	Expression
opeliunxp	|- ( <.x , C >. e. U_x e. A ( {x } X. B) <-> (x e. A /\ C e. B))

Proof of Theorem opeliunxp

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2560	. 2 |- ( <.x , C >. e. U_x e. A ( {x } X. B) -> <.x , C >. e. _V)
2		vex 2554	. . 3 |- x e. _V
3		elex 2560	. . . 4 |- ( C e. B -> C e. _V)
4	3	adantl 262	. . 3 |- ((x e. A /\ C e. B) -> C e. _V)
5		opexgOLD 3956	. . 3 |- ((x e. _V /\ C e. _V) -> <.x , C >. e. _V)
6	2, 4, 5	sylancr 393	. 2 |- ((x e. A /\ C e. B) -> <.x , C >. e. _V)
7		df-rex 2306	. . . . . 6 |- (E.x e. A y e. ( {x } X. B) <-> E.x(x e. A /\ y e. ( {x } X. B)))
8		nfv 1418	. . . . . . 7 |- F/z(x e. A /\ y e. ( {x } X. B))
9		nfs1v 1812	. . . . . . . 8 |- F/x[z / x]x e. A
10		nfcv 2175	. . . . . . . . . 10 |- F/_x {z }
11		nfcsb1v 2876	. . . . . . . . . 10 |- F/_x [_z / x ]_B
12	10, 11	nfxp 4314	. . . . . . . . 9 |- F/_x( {z } X. [_z / x ]_B)
13	12	nfcri 2169	. . . . . . . 8 |- F/x y e. ( {z } X. [_z / x ]_B)
14	9, 13	nfan 1454	. . . . . . 7 |- F/x([z / x]x e. A /\ y e. ( {z } X. [_z / x ]_B))
15		sbequ12 1651	. . . . . . . 8 |- (x = z -> (x e. A <-> [z / x]x e. A))
16		sneq 3378	. . . . . . . . . 10 |- (x = z -> {x } = {z })
17		csbeq1a 2854	. . . . . . . . . 10 |- (x = z -> B = [_z / x ]_B)
18	16, 17	xpeq12d 4313	. . . . . . . . 9 |- (x = z -> ( {x } X. B) = ( {z } X. [_z / x ]_B))
19	18	eleq2d 2104	. . . . . . . 8 |- (x = z -> (y e. ( {x } X. B) <-> y e. ( {z } X. [_z / x ]_B)))
20	15, 19	anbi12d 442	. . . . . . 7 |- (x = z -> ((x e. A /\ y e. ( {x } X. B)) <-> ([z / x]x e. A /\ y e. ( {z } X. [_z / x ]_B))))
21	8, 14, 20	cbvex 1636	. . . . . 6 |- (E.x(x e. A /\ y e. ( {x } X. B)) <-> E.z([z / x]x e. A /\ y e. ( {z } X. [_z / x ]_B)))
22	7, 21	bitri 173	. . . . 5 |- (E.x e. A y e. ( {x } X. B) <-> E.z([z / x]x e. A /\ y e. ( {z } X. [_z / x ]_B)))
23		eleq1 2097	. . . . . . 7 |- (y = <.x , C >. -> (y e. ( {z } X. [_z / x ]_B) <-> <.x , C >. e. ( {z } X. [_z / x ]_B)))
24	23	anbi2d 437	. . . . . 6 |- (y = <.x , C >. -> (([z / x]x e. A /\ y e. ( {z } X. [_z / x ]_B)) <-> ([z / x]x e. A /\ <.x , C >. e. ( {z } X. [_z / x ]_B))))
25	24	exbidv 1703	. . . . 5 |- (y = <.x , C >. -> (E.z([z / x]x e. A /\ y e. ( {z } X. [_z / x ]_B)) <-> E.z([z / x]x e. A /\ <.x , C >. e. ( {z } X. [_z / x ]_B))))
26	22, 25	syl5bb 181	. . . 4 |- (y = <.x , C >. -> (E.x e. A y e. ( {x } X. B) <-> E.z([z / x]x e. A /\ <.x , C >. e. ( {z } X. [_z / x ]_B))))
27		df-iun 3650	. . . 4 |- U_x e. A ( {x } X. B) = {y | E.x e. A y e. ( {x } X. B) }
28	26, 27	elab2g 2683	. . 3 |- ( <.x , C >. e. _V -> ( <.x , C >. e. U_x e. A ( {x } X. B) <-> E.z([z / x]x e. A /\ <.x , C >. e. ( {z } X. [_z / x ]_B))))
29		opelxp 4317	. . . . . . 7 |- ( <.x , C >. e. ( {z } X. [_z / x ]_B) <-> (x e. {z } /\ C e. [_z / x ]_B))
30	29	anbi2i 430	. . . . . 6 |- (([z / x]x e. A /\ <.x , C >. e. ( {z } X. [_z / x ]_B)) <-> ([z / x]x e. A /\ (x e. {z } /\ C e. [_z / x ]_B)))
31		an12 495	. . . . . 6 |- (([z / x]x e. A /\ (x e. {z } /\ C e. [_z / x ]_B)) <-> (x e. {z } /\ ([z / x]x e. A /\ C e. [_z / x ]_B)))
32		elsn 3382	. . . . . . . 8 |- (x e. {z } <-> x = z)
33		equcom 1590	. . . . . . . 8 |- (x = z <-> z = x)
34	32, 33	bitri 173	. . . . . . 7 |- (x e. {z } <-> z = x)
35	34	anbi1i 431	. . . . . 6 |- ((x e. {z } /\ ([z / x]x e. A /\ C e. [_z / x ]_B)) <-> (z = x /\ ([z / x]x e. A /\ C e. [_z / x ]_B)))
36	30, 31, 35	3bitri 195	. . . . 5 |- (([z / x]x e. A /\ <.x , C >. e. ( {z } X. [_z / x ]_B)) <-> (z = x /\ ([z / x]x e. A /\ C e. [_z / x ]_B)))
37	36	exbii 1493	. . . 4 |- (E.z([z / x]x e. A /\ <.x , C >. e. ( {z } X. [_z / x ]_B)) <-> E.z(z = x /\ ([z / x]x e. A /\ C e. [_z / x ]_B)))
38		sbequ12r 1652	. . . . . 6 |- (z = x -> ([z / x]x e. A <-> x e. A))
39	17	equcoms 1591	. . . . . . . 8 |- (z = x -> B = [_z / x ]_B)
40	39	eqcomd 2042	. . . . . . 7 |- (z = x -> [_z / x ]_B = B)
41	40	eleq2d 2104	. . . . . 6 |- (z = x -> ( C e. [_z / x ]_B <-> C e. B))
42	38, 41	anbi12d 442	. . . . 5 |- (z = x -> (([z / x]x e. A /\ C e. [_z / x ]_B) <-> (x e. A /\ C e. B)))
43	2, 42	ceqsexv 2587	. . . 4 |- (E.z(z = x /\ ([z / x]x e. A /\ C e. [_z / x ]_B)) <-> (x e. A /\ C e. B))
44	37, 43	bitri 173	. . 3 |- (E.z([z / x]x e. A /\ <.x , C >. e. ( {z } X. [_z / x ]_B)) <-> (x e. A /\ C e. B))
45	28, 44	syl6bb 185	. 2 |- ( <.x , C >. e. _V -> ( <.x , C >. e. U_x e. A ( {x } X. B) <-> (x e. A /\ C e. B)))
46	1, 6, 45	pm5.21nii 619	1 |- ( <.x , C >. e. U_x e. A ( {x } X. B) <-> (x e. A /\ C e. B))

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1242  E.wex 1378   e. wcel 1390  [wsb 1642  E.wrex 2301   _Vcvv 2551   [_csb 2846   {csn 3367   <.cop 3370   U_ciun 3648   X. cxp 4286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-opab 3810  df-xp 4294

This theorem is referenced by:  eliunxp  4418  opeliunxp2  4419