Theorem opeliun2xp 47008

Description: Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5744. (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 5000	. . 3 ⊢ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) = {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦})}
2	1	eleq2i 2826	. 2 ⊢ (⟨𝐶, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦})})
3		opex 5465	. . 3 ⊢ ⟨𝐶, 𝑦⟩ ∈ V
4		df-rex 3072	. . . . 5 ⊢ (∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 × {𝑦})))
5		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 × {𝑦}))
6		nfs1v 2154	. . . . . . 7 ⊢ Ⅎ𝑦[𝑧 / 𝑦]𝑦 ∈ 𝐵
7		nfcsb1v 3919	. . . . . . . . 9 ⊢ Ⅎ𝑦⦋𝑧 / 𝑦⦌𝐴
8		nfcv 2904	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑧}
9	7, 8	nfxp 5710	. . . . . . . 8 ⊢ Ⅎ𝑦(⦋𝑧 / 𝑦⦌𝐴 × {𝑧})
10	9	nfcri 2891	. . . . . . 7 ⊢ Ⅎ𝑦 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})
11	6, 10	nfan 1903	. . . . . 6 ⊢ Ⅎ𝑦([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}))
12		sbequ12 2244	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ [𝑧 / 𝑦]𝑦 ∈ 𝐵))
13		csbeq1a 3908	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → 𝐴 = ⦋𝑧 / 𝑦⦌𝐴)
14		sneq 4639	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → {𝑦} = {𝑧})
15	13, 14	xpeq12d 5708	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝐴 × {𝑦}) = (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}))
16	15	eleq2d 2820	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑥 ∈ (𝐴 × {𝑦}) ↔ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})))
17	12, 16	anbi12d 632	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 × {𝑦})) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}))))
18	5, 11, 17	cbvexv1 2339	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (𝐴 × {𝑦})) ↔ ∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})))
19	4, 18	bitri 275	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})))
20		eleq1 2822	. . . . . 6 ⊢ (𝑥 = ⟨𝐶, 𝑦⟩ → (𝑥 ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}) ↔ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})))
21	20	anbi2d 630	. . . . 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 1925	. . . 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 283	. . 3 ⊢ (𝑥 = ⟨𝐶, 𝑦⟩ → (∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦}) ↔ ∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}))))
24	3, 23	elab 3669	. 2 ⊢ (⟨𝐶, 𝑦⟩ ∈ {𝑥 ∣ ∃𝑦 ∈ 𝐵 𝑥 ∈ (𝐴 × {𝑦})} ↔ ∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})))
25		opelxp 5713	. . . . . 6 ⊢ (⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧}) ↔ (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ 𝑦 ∈ {𝑧}))
26	25	anbi2i 624	. . . . 5 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ 𝑦 ∈ {𝑧})))
27		an13 646	. . . . . 6 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ 𝑦 ∈ {𝑧})) ↔ (𝑦 ∈ {𝑧} ∧ (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ [𝑧 / 𝑦]𝑦 ∈ 𝐵)))
28		ancom 462	. . . . . . 7 ⊢ ((𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ [𝑧 / 𝑦]𝑦 ∈ 𝐵) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴))
29	28	anbi2i 624	. . . . . 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 275	. . . . 5 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ∧ 𝑦 ∈ {𝑧})) ↔ (𝑦 ∈ {𝑧} ∧ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴)))
31		velsn 4645	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧} ↔ 𝑦 = 𝑧)
32		equcom 2022	. . . . . . 7 ⊢ (𝑦 = 𝑧 ↔ 𝑧 = 𝑦)
33	31, 32	bitri 275	. . . . . 6 ⊢ (𝑦 ∈ {𝑧} ↔ 𝑧 = 𝑦)
34	33	anbi1i 625	. . . . 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 297	. . . 4 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})) ↔ (𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴)))
36	35	exbii 1851	. . 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 2245	. . . . 5 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
38	13	equcoms 2024	. . . . . . 7 ⊢ (𝑧 = 𝑦 → 𝐴 = ⦋𝑧 / 𝑦⦌𝐴)
39	38	eqcomd 2739	. . . . . 6 ⊢ (𝑧 = 𝑦 → ⦋𝑧 / 𝑦⦌𝐴 = 𝐴)
40	39	eleq2d 2820	. . . . 5 ⊢ (𝑧 = 𝑦 → (𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴 ↔ 𝐶 ∈ 𝐴))
41	37, 40	anbi12d 632	. . . 4 ⊢ (𝑧 = 𝑦 → (([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴) ↔ (𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴)))
42	41	equsexvw 2009	. . 3 ⊢ (∃𝑧(𝑧 = 𝑦 ∧ ([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ 𝐶 ∈ ⦋𝑧 / 𝑦⦌𝐴)) ↔ (𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴))
43	36, 42	bitri 275	. 2 ⊢ (∃𝑧([𝑧 / 𝑦]𝑦 ∈ 𝐵 ∧ ⟨𝐶, 𝑦⟩ ∈ (⦋𝑧 / 𝑦⦌𝐴 × {𝑧})) ↔ (𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴))
44	2, 24, 43	3bitri 297	1 ⊢ (⟨𝐶, 𝑦⟩ ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴))

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542  ∃wex 1782  [wsb 2068   ∈ wcel 2107  {cab 2710  ∃wrex 3071  ⦋csb 3894  {csn 4629  ⟨cop 4635  ∪ ciun 4998   × cxp 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iun 5000  df-opab 5212  df-xp 5683

This theorem is referenced by:  eliunxp2  47009