Theorem elxp4 2640

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 2641 and elxp6 3093.

Assertion

Ref	Expression
elxp4	|- (A e. (B X. C) <-> (A = <.U.dom {A}, U.ran {A}>. /\ (U.dom {A} e. B /\ U.ran {A} e. C)))

Proof of Theorem elxp4

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		sneq 1816	. . . . . . . . 9 |- (A = <.x, U.ran {A}>. -> {A} = {<.x, U.ran {A}>.})
26	25	dmeqd 2533	. . . . . . . 8 |- (A = <.x, U.ran {A}>. -> dom {A} = dom {<.x, U.ran {A}>.})
27	26	unieqd 1929	. . . . . . 7 |- (A = <.x, U.ran {A}>. -> U.dom {A} = U.dom {<.x, U.ran {A}>.})
28	5	op1sta 2635	. . . . . . 7 |- U.dom {<.x, U.ran {A}>.} = x
29	27, 28	syl6req 1141	. . . . . 6 |- (A = <.x, U.ran {A}>. -> x = U.dom {A})
30	29	pm4.71ri 484	. . . . 5 |- (A = <.x, U.ran {A}>. <-> (x = U.dom {A} /\ A = <.x, U.ran {A}>.))
31	30	anbi1i 368	. . . 4 |- ((A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C)) <-> ((x = U.dom {A} /\ A = <.x, U.ran {A}>.) /\ (x e. B /\ U.ran {A} e. C)))
32		anass 336	. . . 4 |- (((x = U.dom {A} /\ A = <.x, U.ran {A}>.) /\ (x e. B /\ U.ran {A} e. C)) <-> (x = U.dom {A} /\ (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))))
33	24, 31, 32	3bitr 155	. . 3 |- (E.y(A = <.x, y>. /\ (x e. B /\ y e. C)) <-> (x = U.dom {A} /\ (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))))
34	33	biex 733	. 2 |- (E.xE.y(A = <.x, y>. /\ (x e. B /\ y e. C)) <-> E.x(x = U.dom {A} /\ (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))))
35		dmexg 2551	. . . . 5 |- ({A} e. V -> dom {A} e. V)
36	14, 35	ax-mp 6	. . . 4 |- dom {A} e. V
37	36	uniex 1947	. . 3 |- U.dom {A} e. V
38		opeq1 1876	. . . . 5 |- (x = U.dom {A} -> <.x, U.ran {A}>. = <.U.dom {A}, U.ran {A}>.)
39	38	cleq2d 1112	. . . 4 |- (x = U.dom {A} -> (A = <.x, U.ran {A}>. <-> A = <.U.dom {A}, U.ran {A}>.))
40		eleq1 1149	. . . . 5 |- (x = U.dom {A} -> (x e. B <-> U.dom {A} e. B))
41	40	anbi1d 469	. . . 4 |- (x = U.dom {A} -> ((x e. B /\ U.ran {A} e. C) <-> (U.dom {A} e. B /\ U.ran {A} e. C)))
42	39, 41	anbi12d 476	. . 3 |- (x = U.dom {A} -> ((A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C)) <-> (A = <.U.dom {A}, U.ran {A}>. /\ (U.dom {A} e. B /\ U.ran {A} e. C))))
43	37, 42	ceqsexv 1371	. 2 |- (E.x(x = U.dom {A} /\ (A = <.x, U.ran {A}>. /\ (x e. B /\ U.ran {A} e. C))) <-> (A = <.U.dom {A}, U.ran {A}>. /\ (U.dom {A} e. B /\ U.ran {A} e. C)))
44	1, 34, 43	3bitr 155	1 |- (A e. (B X. C) <-> (A = <.U.dom {A}, U.ran {A}>. /\ (U.dom {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   X. cxp 2408  dom cdm 2410  ran crn 2411

This theorem is referenced by:  elxp6 3093  xpdom2 3345  xpmapenlem3 3393  xpmapenlem5 3395

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-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-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429

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