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 ∈ (B × C) ↔ (A = ⟨∪dom {A}, ∪ran {A}⟩ ∧ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ 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 ∈ (B × C) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ 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⟩ → ∪ran {A} = ∪ran {⟨x, y⟩})
5		vex 2862	. . . . . . . . . 10 ⊢ x ∈ V
6		vex 2862	. . . . . . . . . 10 ⊢ y ∈ V
7	5, 6	op2nda 5076	. . . . . . . . 9 ⊢ ∪ran {⟨x, y⟩} = y
8	4, 7	syl6req 2402	. . . . . . . 8 ⊢ (A = ⟨x, y⟩ → y = ∪ran {A})
9	8	adantr 451	. . . . . . 7 ⊢ ((A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) → y = ∪ran {A})
10	9	pm4.71ri 614	. . . . . 6 ⊢ ((A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ (y = ∪ran {A} ∧ (A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C))))
11	10	exbii 1582	. . . . 5 ⊢ (∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ ∃y(y = ∪ran {A} ∧ (A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C))))
12		snex 4111	. . . . . . . 8 ⊢ {A} ∈ V
13	12	rnex 5107	. . . . . . 7 ⊢ ran {A} ∈ V
14	13	uniex 4317	. . . . . 6 ⊢ ∪ran {A} ∈ V
15		opeq2 4579	. . . . . . . 8 ⊢ (y = ∪ran {A} → ⟨x, y⟩ = ⟨x, ∪ran {A}⟩)
16	15	eqeq2d 2364	. . . . . . 7 ⊢ (y = ∪ran {A} → (A = ⟨x, y⟩ ↔ A = ⟨x, ∪ran {A}⟩))
17		eleq1 2413	. . . . . . . 8 ⊢ (y = ∪ran {A} → (y ∈ C ↔ ∪ran {A} ∈ C))
18	17	anbi2d 684	. . . . . . 7 ⊢ (y = ∪ran {A} → ((x ∈ B ∧ y ∈ C) ↔ (x ∈ B ∧ ∪ran {A} ∈ C)))
19	16, 18	anbi12d 691	. . . . . 6 ⊢ (y = ∪ran {A} → ((A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C))))
20	14, 19	ceqsexv 2894	. . . . 5 ⊢ (∃y(y = ∪ran {A} ∧ (A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C))) ↔ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C)))
21	11, 20	bitri 240	. . . 4 ⊢ (∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C)))
22		sneq 3744	. . . . . . . . 9 ⊢ (A = ⟨x, ∪ran {A}⟩ → {A} = {⟨x, ∪ran {A}⟩})
23	22	dmeqd 4909	. . . . . . . 8 ⊢ (A = ⟨x, ∪ran {A}⟩ → dom {A} = dom {⟨x, ∪ran {A}⟩})
24	23	unieqd 3902	. . . . . . 7 ⊢ (A = ⟨x, ∪ran {A}⟩ → ∪dom {A} = ∪dom {⟨x, ∪ran {A}⟩})
25	5, 14	op1sta 5072	. . . . . . 7 ⊢ ∪dom {⟨x, ∪ran {A}⟩} = x
26	24, 25	syl6req 2402	. . . . . 6 ⊢ (A = ⟨x, ∪ran {A}⟩ → x = ∪dom {A})
27	26	pm4.71ri 614	. . . . 5 ⊢ (A = ⟨x, ∪ran {A}⟩ ↔ (x = ∪dom {A} ∧ A = ⟨x, ∪ran {A}⟩))
28	27	anbi1i 676	. . . 4 ⊢ ((A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C)) ↔ ((x = ∪dom {A} ∧ A = ⟨x, ∪ran {A}⟩) ∧ (x ∈ B ∧ ∪ran {A} ∈ C)))
29		anass 630	. . . 4 ⊢ (((x = ∪dom {A} ∧ A = ⟨x, ∪ran {A}⟩) ∧ (x ∈ B ∧ ∪ran {A} ∈ C)) ↔ (x = ∪dom {A} ∧ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C))))
30	21, 28, 29	3bitri 262	. . 3 ⊢ (∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ (x = ∪dom {A} ∧ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C))))
31	30	exbii 1582	. 2 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ ∃x(x = ∪dom {A} ∧ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C))))
32	12	dmex 5106	. . . 4 ⊢ dom {A} ∈ V
33	32	uniex 4317	. . 3 ⊢ ∪dom {A} ∈ V
34		opeq1 4578	. . . . 5 ⊢ (x = ∪dom {A} → ⟨x, ∪ran {A}⟩ = ⟨∪dom {A}, ∪ran {A}⟩)
35	34	eqeq2d 2364	. . . 4 ⊢ (x = ∪dom {A} → (A = ⟨x, ∪ran {A}⟩ ↔ A = ⟨∪dom {A}, ∪ran {A}⟩))
36		eleq1 2413	. . . . 5 ⊢ (x = ∪dom {A} → (x ∈ B ↔ ∪dom {A} ∈ B))
37	36	anbi1d 685	. . . 4 ⊢ (x = ∪dom {A} → ((x ∈ B ∧ ∪ran {A} ∈ C) ↔ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C)))
38	35, 37	anbi12d 691	. . 3 ⊢ (x = ∪dom {A} → ((A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C)) ↔ (A = ⟨∪dom {A}, ∪ran {A}⟩ ∧ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C))))
39	33, 38	ceqsexv 2894	. 2 ⊢ (∃x(x = ∪dom {A} ∧ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C))) ↔ (A = ⟨∪dom {A}, ∪ran {A}⟩ ∧ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C)))
40	1, 31, 39	3bitri 262	1 ⊢ (A ∈ (B × C) ↔ (A = ⟨∪dom {A}, ∪ran {A}⟩ ∧ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C)))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {csn 3737  ∪cuni 3891  ⟨cop 4561   × 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)