Theorem opeliun2xp 47096

Description: Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5742. (Contributed by AV, 30-Mar-2019.)

Assertion

Ref	Expression
opeliun2xp	⊢ (⟨𝐶, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴))

Proof of Theorem opeliun2xp

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4998	. . 3 ⊢ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) = {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦})}
2	1	eleq2i 2823	. 2 ⊢ (⟨𝐶, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦})})
3		opex 5463	. . 3 ⊢ ⟨𝐶, 𝑦⟩ ∈ V
4		df-rex 3069	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 × {𝑦})))
5		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 × {𝑦}))
6		nfs1v 2151	. . . . . . 7 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝑦 ∈ 𝐵
7		nfcsb1v 3917	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐴
8		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑧}
9	7, 8	nfxp 5708	. . . . . . . 8 ⊢ Ⅎ𝑦(⦋𝑧 / 𝑦⦌𝐴 × {𝑧})
10	9	nfcri 2888	. . . . . . 7 ⊢ Ⅎ𝑦 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})
11	6, 10	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑦([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}))
12		sbequ12 2241	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ [𝑧 / 𝑦]𝑦 ∈ 𝐵))
13		csbeq1a 3906	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑦⦌𝐴)
14		sneq 4637	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
15	13, 14	xpeq12d 5706	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝐴 × {𝑦}) = (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}))
16	15	eleq2d 2817	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ (𝐴 × {𝑦}) ↔ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})))
17	12, 16	anbi12d 629	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 × {𝑦})) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}))))
18	5, 11, 17	cbvexv1 2336	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 × {𝑦})) ↔ ∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})))
19	4, 18	bitri 274	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})))
20		eleq1 2819	. . . . . 6 ⊢ (𝑥 = ⟨𝐶, 𝑦⟩ → (𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}) ↔ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})))
21	20	anbi2d 627	. . . . 5 ⊢ (𝑥 = ⟨𝐶, 𝑦⟩ → (([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}))))
22	21	exbidv 1922	. . . 4 ⊢ (𝑥 = ⟨𝐶, 𝑦⟩ → (∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})) ↔ ∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}))))
23	19, 22	bitrid 282	. . 3 ⊢ (𝑥 = ⟨𝐶, 𝑦⟩ → (∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}))))
24	3, 23	elab 3667	. 2 ⊢ (⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦})} ↔ ∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})))
25		opelxp 5711	. . . . . 6 ⊢ (⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}) ↔ (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ 𝑦 ∈ {𝑧}))
26	25	anbi2i 621	. . . . 5 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ 𝑦 ∈ {𝑧})))
27		an13 643	. . . . . 6 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ 𝑦 ∈ {𝑧})) ↔ (𝑦 ∈ {𝑧} ∧ (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ [𝑧 / 𝑦]𝑦 ∈ 𝐵)))
28		ancom 459	. . . . . . 7 ⊢ ((𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ [𝑧 / 𝑦]𝑦 ∈ 𝐵) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴))
29	28	anbi2i 621	. . . . . 6 ⊢ ((𝑦 ∈ {𝑧} ∧ (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ [𝑧 / 𝑦]𝑦 ∈ 𝐵)) ↔ (𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴)))
30	27, 29	bitri 274	. . . . 5 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ 𝑦 ∈ {𝑧})) ↔ (𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴)))
31		velsn 4643	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
32		equcom 2019	. . . . . . 7 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
33	31, 32	bitri 274	. . . . . 6 ⊢ (𝑦 ∈ {𝑧} ↔ 𝑧 = 𝑦)
34	33	anbi1i 622	. . . . 5 ⊢ ((𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴)) ↔ (𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴)))
35	26, 30, 34	3bitri 296	. . . 4 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})) ↔ (𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴)))
36	35	exbii 1848	. . 3 ⊢ (∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})) ↔ ∃𝑧(𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴)))
37		sbequ12r 2242	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
38	13	equcoms 2021	. . . . . . 7 ⊢ (𝑧 = 𝑦 → 𝐴 = ⦋𝑧 / 𝑦⦌𝐴)
39	38	eqcomd 2736	. . . . . 6 ⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑦⦌𝐴 = 𝐴)
40	39	eleq2d 2817	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ↔ 𝐶 ∈ 𝐴))
41	37, 40	anbi12d 629	. . . 4 ⊢ (𝑧 = 𝑦 → (([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴) ↔ (𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴)))
42	41	equsexvw 2006	. . 3 ⊢ (∃𝑧(𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴))
43	36, 42	bitri 274	. 2 ⊢ (∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})) ↔ (𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴))
44	2, 24, 43	3bitri 296	1 ⊢ (⟨𝐶, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1539  ∃wex 1779  [wsb 2065   ∈ wcel 2104  {cab 2707  ∃wrex 3068  ⦋csb 3892  {csn 4627  ⟨cop 4633  ∪ ciun 4996   × cxp 5673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-iun 4998  df-opab 5210  df-xp 5681

This theorem is referenced by:  eliunxp2  47097