Theorem xpiindim 4761

Description: Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)

Assertion

Ref	Expression
xpiindim	⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

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

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem xpiindim

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

Step	Hyp	Ref	Expression
1		relxp 4733	. . . . . 6 ⊢ Rel (𝐶 × 𝐵)
2	1	rgenw 2532	. . . . 5 ⊢ ∀𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵)
3		r19.2m 3509	. . . . 5 ⊢ ((∃𝑦 𝑦 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵)) → ∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵))
4	2, 3	mpan2 425	. . . 4 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵))
5		reliin 4746	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 Rel (𝐶 × 𝐵) → Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
6	4, 5	syl 14	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
7		relxp 4733	. . 3 ⊢ Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵)
8	6, 7	jctil 312	. 2 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ∧ Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
9		eleq1w 2238	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
10	9	cbvexv 1918	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 ↔ ∃𝑦 𝑦 ∈ 𝐴)
11		r19.28mv 3515	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵) ↔ (𝑤 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)))
12	10, 11	sylbir 135	. . . . . 6 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵) ↔ (𝑤 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)))
13	12	bicomd 141	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝑤 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵)))
14		eliin 3890	. . . . . . 7 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
15	14	elv 2741	. . . . . 6 ⊢ (𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
16	15	anbi2i 457	. . . . 5 ⊢ ((𝑤 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑤 ∈ 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵))
17		opelxp 4654	. . . . . 6 ⊢ (⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵))
18	17	ralbii 2483	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ 𝐵))
19	13, 16, 18	3bitr4g 223	. . . 4 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → ((𝑤 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵)))
20		opelxp 4654	. . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑤 ∈ 𝐶 ∧ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝐵))
21		vex 2740	. . . . . 6 ⊢ 𝑤 ∈ V
22		vex 2740	. . . . . 6 ⊢ 𝑧 ∈ V
23	21, 22	opex 4227	. . . . 5 ⊢ ⟨𝑤, 𝑧⟩ ∈ V
24		eliin 3890	. . . . 5 ⊢ (⟨𝑤, 𝑧⟩ ∈ V → (⟨𝑤, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵)))
25	23, 24	ax-mp 5	. . . 4 ⊢ (⟨𝑤, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵) ↔ ∀𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ (𝐶 × 𝐵))
26	19, 20, 25	3bitr4g 223	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (⟨𝑤, 𝑧⟩ ∈ (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ↔ ⟨𝑤, 𝑧⟩ ∈ ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)))
27	26	eqrelrdv2 4723	. 2 ⊢ (((Rel (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) ∧ Rel ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵)) ∧ ∃𝑦 𝑦 ∈ 𝐴) → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))
28	8, 27	mpancom 422	1 ⊢ (∃𝑦 𝑦 ∈ 𝐴 → (𝐶 × ∩ 𝑥 ∈ 𝐴 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐶 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  Vcvv 2737  ⟨cop 3595  ∩ ciin 3886   × cxp 4622  Rel wrel 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-iin 3888  df-opab 4063  df-xp 4630  df-rel 4631

This theorem is referenced by:  xpriindim  4762