Theorem opeliunxp2 5780

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 5073	. . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
2		relxp 5636	. . . . . 6 ⊢ Rel ({𝑥} × 𝐵)
3	2	rgenw 3057	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵)
4		reliun 5759	. . . . 5 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵))
5	3, 4	mpbir 232	. . . 4 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
6	5	brrelex1i 5674	. . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 → 𝐶 ∈ V)
7	1, 6	sylbir 236	. 2 ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V)
8		elex 3452	. . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V)
9	8	adantr 481	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) → 𝐶 ∈ V)
10		nfiu1 4957	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
11	10	nfel2 2919	. . . 4 ⊢ Ⅎ𝑥⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
12		nfv 1921	. . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)
13	11, 12	nfbi 1910	. . 3 ⊢ Ⅎ𝑥(⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))
14		opeq1 4804	. . . . 5 ⊢ (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
15	14	eleq1d 2824	. . . 4 ⊢ (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
16		eleq1 2827	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
17		opeliunxp2.1	. . . . . 6 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸)
18	17	eleq2d 2825	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐷 ∈ 𝐵 ↔ 𝐷 ∈ 𝐸))
19	16, 18	anbi12d 638	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)))
20	15, 19	bibi12d 346	. . 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 5685	. . 3 ⊢ (⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
22	13, 20, 21	vtoclg1f 3514	. 2 ⊢ (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)))
23	7, 9, 22	pm5.21nii 379	1 ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  Vcvv 3431  {csn 4555  ⟨cop 4561  ∪ ciun 4921   class class class wbr 5072   × cxp 5616  Rel wrel 5623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-iun 4923  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625

This theorem is referenced by:  mpoxopn0yelv  8153  mpoxopxnop0  8155  eldmcoa  18023  dmdprd  19966  ply1frcl  22304  cnextfres  24052  eldv  25883  perfdvf  25888  eltayl  26343  dfcnv2  32767  gsumwrd2dccat  33159  cvmliftlem1  35513  filnetlem3  36608