Theorem elxp5 2641

Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 2640 when the double intersection does not create class existence problems (caused by int0 1978).

Assertion

Ref	Expression
elxp5	|- (A e. (B X. C) <-> (A = <.|^||^|A, U.ran {A}>. /\ (|^||^|A e. B /\ U.ran {A} e. C)))

Proof of Theorem elxp5

Step	Hyp	Ref	Expression
1		elxp 2442	. 2 |- (A e. (B X. C) <-> E.xE.y(A = <.x, y>. /\ (x e. B /\ y e. C)))
2		sneq 1816	. . . . . . . . . . . 12 |- (A = <.x, y>. -> {A} = {<.x, y>.})
3	2	rneqd 2557	. . . . . . . . . . 11 |- (A = <.x, y>. -> ran {A} = ran {<.x, y>.})
4	3	unieqd 1929	. . . . . . . . . 10 |- (A = <.x, y>. -> U.ran {A} = U.ran {<.x, y>.})
5		visset 1350	. . . . . . . . . . 11 |- x e. V
6		visset 1350	. . . . . . . . . . 11 |- y e. V
7	5, 6	op2nda 2639	. . . . . . . . . 10 |- U.ran {<.x, y>.} = y
8	4, 7	syl6req 1141	. . . . . . . . 9 |- (A = <.x, y>. -> y = U.ran {A})
9	8	pm4.71ri 484	. . . . . . . 8 |- (A = <.x, y>. <-> (y = U.ran {A} /\ A = <.x, y>.))
10	9	anbi1i 368	. . . . . . 7 |- ((A = <.x, y>. /\ (x e. B /\ y e. C)) <-> ((y = U.ran {A} /\ A = <.x, y>.) /\ (x e. B /\ y e. C)))
11		anass 336	. . . . . . 7 |- (((y = U.ran {A} /\ A = <.x, y>.) /\ (x e. B /\ y e. C)) <-> (y = U.ran {A} /\ (A = <.x, y>. /\ (x e. B /\ y e. C))))
12	10, 11	bitr 151	. . . . . 6 |- ((A = <.x, y>. /\ (x e. B /\ y e. C)) <-> (y = U.ran {A} /\ (A = <.x, y>. /\ (x e. B /\ y e. C))))
13	12	biex 733	. . . . 5 |- (E.y(A = <.x, y>. /\ (x e. B /\ y e. C)) <-> E.y(y = U.ran {A} /\ (A = <.x, y>. /\ (x e. B /\ y e. C))))
14		snex 1859	. . . . . . . 8 |- {A} e. V
15		rnexg 2569	. . . . . . . 8 |- ({A} e. V -> ran {A} e. V)
16	14, 15	ax-mp 6	. . . . . . 7 |- ran {A} e. V
17	16	uniex 1947	. . . . . 6 |- U.ran {A} e. V
18		opeq2 1877	. . . . . . . 8 |- (y = U.ran {A} -> <.x, y>. = <.x, U.ran {A}>.)
19	18	cleq2d 1112	. . . . . . 7 |- (y = U.ran {A} -> (A = <.x, y>. <-> A = <.x, U.ran {A}>.))
20		eleq1 1149	. . . . . . . 8 |- (y = U.ran {A} -> (y e. C <-> U.ran {A} e. C))
21	20	anbi2d 468	. . . . . . 7 |- (y = U.ran {A} -> ((x e. B /\ y e. C) <-> (x e. B /\ U.ran {A} e. C)))
22	19, 21	anbi12d 476	. . . . . 6 |- (y = U.ran {A} -> ((A = <.x, y>. /\ (x e. B /\ y e. C)) <-> (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))))
23	17, 22	ceqsexv 1371	. . . . 5 |- (E.y(y = U.ran {A} /\ (A = <.x, y>. /\ (x e. B /\ y e. C))) <-> (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C)))
24	13, 23	bitr 151	. . . 4 |- (E.y(A = <.x, y>. /\ (x e. B /\ y e. C)) <-> (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C)))
25		inteq 1968	. . . . . . . 8 |- (A = <.x, U.ran {A}>. -> |^|A = |^|<.x, U.ran {A}>.)
26	25	inteqd 1970	. . . . . . 7 |- (A = <.x, U.ran {A}>. -> |^||^|A = |^||^|<.x, U.ran {A}>.)
27	5	op1stb 1992	. . . . . . 7 |- |^||^|<.x, U.ran {A}>. = x
28	26, 27	syl6req 1141	. . . . . 6 |- (A = <.x, U.ran {A}>. -> x = |^||^|A)
29	28	pm4.71ri 484	. . . . 5 |- (A = <.x, U.ran {A}>. <-> (x = |^||^|A /\ A = <.x, U.ran {A}>.))
30	29	anbi1i 368	. . . 4 |- ((A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C)) <-> ((x = |^||^|A /\ A = <.x, U.ran {A}>.) /\ (x e. B /\ U.ran {A} e. C)))
31		anass 336	. . . 4 |- (((x = |^||^|A /\ A = <.x, U.ran {A}>.) /\ (x e. B /\ U.ran {A} e. C)) <-> (x = |^||^|A /\ (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))))
32	24, 30, 31	3bitr 155	. . 3 |- (E.y(A = <.x, y>. /\ (x e. B /\ y e. C)) <-> (x = |^||^|A /\ (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))))
33	32	biex 733	. 2 |- (E.xE.y(A = <.x, y>. /\ (x e. B /\ y e. C)) <-> E.x(x = |^||^|A /\ (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))))
34		eleq1 1149	. . . . . 6 |- (x = |^||^|A -> (x e. V <-> |^||^|A e. V))
35	5, 34	mpbii 168	. . . . 5 |- (x = |^||^|A -> |^||^|A e. V)
36	35	adantr 306	. . . 4 |- ((x = |^||^|A /\ (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))) -> |^||^|A e. V)
37	36	19.23aiv 952	. . 3 |- (E.x(x = |^||^|A /\ (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))) -> |^||^|A e. V)
38		elisset 1354	. . . . 5 |- (|^||^|A e. B -> |^||^|A e. V)
39	38	adantr 306	. . . 4 |- ((|^||^|A e. B /\ U.ran {A} e. C) -> |^||^|A e. V)
40	39	adantl 305	. . 3 |- ((A = <.|^||^|A, U.ran {A}>. /\ (|^||^|A e. B /\ U.ran {A} e. C)) -> |^||^|A e. V)
41		opeq1 1876	. . . . . 6 |- (x = |^||^|A -> <.x, U.ran {A}>. = <.|^||^|A, U.ran {A}>.)
42	41	cleq2d 1112	. . . . 5 |- (x = |^||^|A -> (A = <.x, U.ran {A}>. <-> A = <.|^||^|A, U.ran {A}>.))
43		eleq1 1149	. . . . . 6 |- (x = |^||^|A -> (x e. B <-> |^||^|A e. B))
44	43	anbi1d 469	. . . . 5 |- (x = |^||^|A -> ((x e. B /\ U.ran {A} e. C) <-> (|^||^|A e. B /\ U.ran {A} e. C)))
45	42, 44	anbi12d 476	. . . 4 |- (x = |^||^|A -> ((A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C)) <-> (A = <.|^||^|A, U.ran {A}>. /\ (|^||^|A e. B /\ U.ran {A} e. C))))
46	45	ceqsexgv 1412	. . 3 |- (|^||^|A e. V -> (E.x(x = |^||^|A /\ (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))) <-> (A = <.|^||^|A, U.ran {A}>. /\ (|^||^|A e. B /\ U.ran {A} e. C))))
47	37, 40, 46	pm5.21nii 504	. 2 |- (E.x(x = |^||^|A /\ (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))) <-> (A = <.|^||^|A, U.ran {A}>. /\ (|^||^|A e. B /\ U.ran {A} e. C)))
48	1, 33, 47	3bitr 155	1 |- (A e. (B X. C) <-> (A = <.|^||^|A, U.ran {A}>. /\ (|^||^|A e. B /\ U.ran {A} e. C)))

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  Vcvv 1348  {csn 1808  <.cop 1810  U.cuni 1919  |^|cint 1965   X. cxp 2408  ran crn 2411

This theorem is referenced by:  mapunen 3397  xpnnen 4927

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-int 1966  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429

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