Theorem opeliunxp2 4822

Description: Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 14-Feb-2015.)

Hypothesis

Ref	Expression
opeliunxp2.1	⊢ (x = C → B = E)

Assertion

Ref	Expression
opeliunxp2	⊢ (⟨C, D⟩ ∈ ∪x ∈ A ({x} × B) ↔ (C ∈ A ∧ D ∈ E))

Distinct variable groups:   x,C   x,D   x,E   x,A

Allowed substitution hint:   B(x)

Proof of Theorem opeliunxp2

Step	Hyp	Ref	Expression
1		elex 2867	. . . 4 ⊢ (⟨C, D⟩ ∈ ∪x ∈ A ({x} × B) → ⟨C, D⟩ ∈ V)
2		opexb 4603	. . . 4 ⊢ (⟨C, D⟩ ∈ V ↔ (C ∈ V ∧ D ∈ V))
3	1, 2	sylib 188	. . 3 ⊢ (⟨C, D⟩ ∈ ∪x ∈ A ({x} × B) → (C ∈ V ∧ D ∈ V))
4	3	simpld 445	. 2 ⊢ (⟨C, D⟩ ∈ ∪x ∈ A ({x} × B) → C ∈ V)
5		elex 2867	. . 3 ⊢ (C ∈ A → C ∈ V)
6	5	adantr 451	. 2 ⊢ ((C ∈ A ∧ D ∈ E) → C ∈ V)
7		nfcv 2489	. . 3 ⊢ ℲxC
8		nfiu1 3997	. . . . 5 ⊢ Ⅎx∪x ∈ A ({x} × B)
9	8	nfel2 2501	. . . 4 ⊢ Ⅎx⟨C, D⟩ ∈ ∪x ∈ A ({x} × B)
10		nfv 1619	. . . 4 ⊢ Ⅎx(C ∈ A ∧ D ∈ E)
11	9, 10	nfbi 1834	. . 3 ⊢ Ⅎx(⟨C, D⟩ ∈ ∪x ∈ A ({x} × B) ↔ (C ∈ A ∧ D ∈ E))
12		opeq1 4578	. . . . 5 ⊢ (x = C → ⟨x, D⟩ = ⟨C, D⟩)
13	12	eleq1d 2419	. . . 4 ⊢ (x = C → (⟨x, D⟩ ∈ ∪x ∈ A ({x} × B) ↔ ⟨C, D⟩ ∈ ∪x ∈ A ({x} × B)))
14		eleq1 2413	. . . . 5 ⊢ (x = C → (x ∈ A ↔ C ∈ A))
15		opeliunxp2.1	. . . . . 6 ⊢ (x = C → B = E)
16	15	eleq2d 2420	. . . . 5 ⊢ (x = C → (D ∈ B ↔ D ∈ E))
17	14, 16	anbi12d 691	. . . 4 ⊢ (x = C → ((x ∈ A ∧ D ∈ B) ↔ (C ∈ A ∧ D ∈ E)))
18	13, 17	bibi12d 312	. . 3 ⊢ (x = C → ((⟨x, D⟩ ∈ ∪x ∈ A ({x} × B) ↔ (x ∈ A ∧ D ∈ B)) ↔ (⟨C, D⟩ ∈ ∪x ∈ A ({x} × B) ↔ (C ∈ A ∧ D ∈ E))))
19		opeliunxp 4820	. . 3 ⊢ (⟨x, D⟩ ∈ ∪x ∈ A ({x} × B) ↔ (x ∈ A ∧ D ∈ B))
20	7, 11, 18, 19	vtoclgf 2913	. 2 ⊢ (C ∈ V → (⟨C, D⟩ ∈ ∪x ∈ A ({x} × B) ↔ (C ∈ A ∧ D ∈ E)))
21	4, 6, 20	pm5.21nii 342	1 ⊢ (⟨C, D⟩ ∈ ∪x ∈ A ({x} × B) ↔ (C ∈ A ∧ D ∈ E))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737  ∪ciun 3969  ⟨cop 4561   × cxp 4770

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-csb 3137  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-iun 3971  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-xp 4784

This theorem is referenced by: (None)