Theorem opeliunxp2 5863

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		df-br 5167	. . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
2		relxp 5718	. . . . . 6 ⊢ Rel ({𝑥} × 𝐵)
3	2	rgenw 3071	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵)
4		reliun 5840	. . . . 5 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵))
5	3, 4	mpbir 231	. . . 4 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
6	5	brrelex1i 5756	. . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 → 𝐶 ∈ V)
7	1, 6	sylbir 235	. 2 ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V)
8		elex 3509	. . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V)
9	8	adantr 480	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) → 𝐶 ∈ V)
10		nfiu1 5050	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
11	10	nfel2 2927	. . . 4 ⊢ Ⅎ𝑥⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
12		nfv 1913	. . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)
13	11, 12	nfbi 1902	. . 3 ⊢ Ⅎ𝑥(⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))
14		opeq1 4897	. . . . 5 ⊢ (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
15	14	eleq1d 2829	. . . 4 ⊢ (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
16		eleq1 2832	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
17		opeliunxp2.1	. . . . . 6 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸)
18	17	eleq2d 2830	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐷 ∈ 𝐵 ↔ 𝐷 ∈ 𝐸))
19	16, 18	anbi12d 631	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)))
20	15, 19	bibi12d 345	. . 3 ⊢ (𝑥 = 𝐶 → ((⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) ↔ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))))
21		opeliunxp 5767	. . 3 ⊢ (⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
22	13, 20, 21	vtoclg1f 3582	. 2 ⊢ (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)))
23	7, 9, 22	pm5.21nii 378	1 ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  {csn 4648  ⟨cop 4654  ∪ ciun 5015   class class class wbr 5166   × cxp 5698  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-iun 5017  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707

This theorem is referenced by:  mpoxopn0yelv  8254  mpoxopxnop0  8256  eldmcoa  18132  dmdprd  20042  ply1frcl  22343  cnextfres  24098  eldv  25953  perfdvf  25958  eltayl  26419  dfcnv2  32694  cvmliftlem1  35253  filnetlem3  36346