Theorem elxp4 5108

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. (Contributed by set.mm contributors, 17-Feb-2004.)

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

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

Step	Hyp	Ref	Expression
1		elxp 4801	. 2 |- (A e. (B X. C) <-> E.xE.y(A = <.x, y>. /\ (x e. B /\ y e. C)))
2		sneq 3744	. . . . . . . . . . 11 |- (A = <.x, y>. -> {A} = {<.x, y>.})
3	2	rneqd 4958	. . . . . . . . . 10 |- (A = <.x, y>. -> ran {A} = ran {<.x, y>.})
4	3	unieqd 3902	. . . . . . . . 9 |- (A = <.x, y>. -> U.ran {A} = U.ran {<.x, y>.})
5		vex 2862	. . . . . . . . . 10 |- x e. _V
6		vex 2862	. . . . . . . . . 10 |- y e. _V
7	5, 6	op2nda 5076	. . . . . . . . 9 |- U.ran {<.x, y>.} = y
8	4, 7	syl6req 2402	. . . . . . . 8 |- (A = <.x, y>. -> y = U.ran {A})
9	8	adantr 451	. . . . . . 7 |- ((A = <.x, y>. /\ (x e. B /\ y e. C)) -> y = U.ran {A})
10	9	pm4.71ri 614	. . . . . 6 |- ((A = <.x, y>. /\ (x e. B /\ y e. C)) <-> (y = U.ran {A} /\ (A = <.x, y>. /\ (x e. B /\ y e. C))))
11	10	exbii 1582	. . . . 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))))
12		snex 4111	. . . . . . . 8 |- {A} e. _V
13	12	rnex 5107	. . . . . . 7 |- ran {A} e. _V
14	13	uniex 4317	. . . . . 6 |- U.ran {A} e. _V
15		opeq2 4579	. . . . . . . 8 |- (y = U.ran {A} -> <.x, y>. = <.x, U.ran {A}>.)
16	15	eqeq2d 2364	. . . . . . 7 |- (y = U.ran {A} -> (A = <.x, y>. <-> A = <.x, U.ran {A}>.))
17		eleq1 2413	. . . . . . . 8 |- (y = U.ran {A} -> (y e. C <-> U.ran {A} e. C))
18	17	anbi2d 684	. . . . . . 7 |- (y = U.ran {A} -> ((x e. B /\ y e. C) <-> (x e. B /\ U.ran {A} e. C)))
19	16, 18	anbi12d 691	. . . . . 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))))
20	14, 19	ceqsexv 2894	. . . . 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)))
21	11, 20	bitri 240	. . . 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)))
22		sneq 3744	. . . . . . . . 9 |- (A = <.x, U.ran {A}>. -> {A} = {<.x, U.ran {A}>.})
23	22	dmeqd 4909	. . . . . . . 8 |- (A = <.x, U.ran {A}>. -> dom {A} = dom {<.x, U.ran {A}>.})
24	23	unieqd 3902	. . . . . . 7 |- (A = <.x, U.ran {A}>. -> U.dom {A} = U.dom {<.x, U.ran {A}>.})
25	5, 14	op1sta 5072	. . . . . . 7 |- U.dom {<.x, U.ran {A}>.} = x
26	24, 25	syl6req 2402	. . . . . 6 |- (A = <.x, U.ran {A}>. -> x = U.dom {A})
27	26	pm4.71ri 614	. . . . 5 |- (A = <.x, U.ran {A}>. <-> (x = U.dom {A} /\ A = <.x, U.ran {A}>.))
28	27	anbi1i 676	. . . 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)))
29		anass 630	. . . 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))))
30	21, 28, 29	3bitri 262	. . 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))))
31	30	exbii 1582	. 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))))
32	12	dmex 5106	. . . 4 |- dom {A} e. _V
33	32	uniex 4317	. . 3 |- U.dom {A} e. _V
34		opeq1 4578	. . . . 5 |- (x = U.dom {A} -> <.x, U.ran {A}>. = <.U.dom {A}, U.ran {A}>.)
35	34	eqeq2d 2364	. . . 4 |- (x = U.dom {A} -> (A = <.x, U.ran {A}>. <-> A = <.U.dom {A}, U.ran {A}>.))
36		eleq1 2413	. . . . 5 |- (x = U.dom {A} -> (x e. B <-> U.dom {A} e. B))
37	36	anbi1d 685	. . . 4 |- (x = U.dom {A} -> ((x e. B /\ U.ran {A} e. C) <-> (U.dom {A} e. B /\ U.ran {A} e. C)))
38	35, 37	anbi12d 691	. . 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))))
39	33, 38	ceqsexv 2894	. 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)))
40	1, 31, 39	3bitri 262	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 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {csn 3737  U.cuni 3891  <.cop 4561   X. cxp 4770  dom cdm 4772  ran crn 4773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-swap 4724  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787

This theorem is referenced by: (None)