Theorem opeliunxp2f 8220

Description: Membership in a union of Cartesian products, using bound-variable hypothesis for 𝐸 instead of distinct variable conditions as in opeliunxp2 5815. (Contributed by AV, 25-Oct-2020.)

Hypotheses

Ref	Expression
opeliunxp2f.f	⊢ Ⅎ𝑥𝐸
opeliunxp2f.e	⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸)

Assertion

Ref	Expression
opeliunxp2f	⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))

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

Allowed substitution hints:   𝐵(𝑥)   𝐸(𝑥)

Proof of Theorem opeliunxp2f

Step	Hyp	Ref	Expression
1		df-br 5104	. . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵))
2		relxp 5669	. . . . . 6 ⊢ Rel ({𝑥} × 𝐵)
3	2	rgenw 3081	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵)
4		reliun 5794	. . . . 5 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × 𝐵))
5	3, 4	mpbir 234	. . . 4 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
6	5	brrelex1i 5707	. . 3 ⊢ (𝐶∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)𝐷 → 𝐶 ∈ V)
7	1, 6	sylbir 238	. 2 ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V)
8		elex 3472	. . 3 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ V)
9	8	adantr 486	. 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸) → 𝐶 ∈ V)
10		nfiu1 4986	. . . . 5 ⊢ Ⅎ𝑥∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
11	10	nfel2 2941	. . . 4 ⊢ Ⅎ𝑥⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)
12		nfv 1947	. . . . 5 ⊢ Ⅎ𝑥 𝐶 ∈ 𝐴
13		opeliunxp2f.f	. . . . . 6 ⊢ Ⅎ𝑥𝐸
14	13	nfel2 2941	. . . . 5 ⊢ Ⅎ𝑥 𝐷 ∈ 𝐸
15	12, 14	nfan 1932	. . . 4 ⊢ Ⅎ𝑥(𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)
16	11, 15	nfbi 1936	. . 3 ⊢ Ⅎ𝑥(⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))
17		opeq1 4833	. . . . 5 ⊢ (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
18	17	eleq1d 2846	. . . 4 ⊢ (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵)))
19		eleq1 2849	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝐶 ∈ 𝐴))
20		opeliunxp2f.e	. . . . . 6 ⊢ (𝑥 = 𝐶 → 𝐵 = 𝐸)
21	20	eleq2d 2847	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝐷 ∈ 𝐵 ↔ 𝐷 ∈ 𝐸))
22	19, 21	anbi12d 644	. . . 4 ⊢ (𝑥 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)))
23	18, 22	bibi12d 348	. . 3 ⊢ (𝑥 = 𝐶 → ((⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵)) ↔ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))))
24		opeliunxp 5718	. . 3 ⊢ (⟨𝑥, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝐷 ∈ 𝐵))
25	16, 23, 24	vtoclg1f 3531	. 2 ⊢ (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸)))
26	7, 9, 25	pm5.21nii 381	1 ⊢ (⟨𝐶, 𝐷⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × 𝐵) ↔ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐸))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2908  ∀wral 3077  Vcvv 3451  {csn 4584  ⟨cop 4590  ∪ ciun 4951   class class class wbr 5103   × cxp 5649  Rel wrel 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-iun 4953  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658

This theorem is used by:  mpoxeldm  8221  fsumcom2  15933  fprodcom2  16144  iunsnima2  33206