Theorem ixpiin 8901

Description: The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.)

Assertion

Ref	Expression
ixpiin	⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶)

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

Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem ixpiin

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.28zv 4472	. . . 4 ⊢ (𝐵 ≠ ∅ → (∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)))
2		eliin 4968	. . . . . 6 ⊢ (𝑓 ∈ V → (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶))
3	2	elv 3460	. . . . 5 ⊢ (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶)
4		vex 3459	. . . . . . 7 ⊢ 𝑓 ∈ V
5	4	elixp 8881	. . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
6	5	ralbii 3077	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
7	3, 6	bitri 275	. . . 4 ⊢ (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
8	4	elixp 8881	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶))
9		fvex 6878	. . . . . . . . 9 ⊢ (𝑓‘𝑥) ∈ V
10		eliin 4968	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) ∈ V → ((𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶))
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ ((𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)
12	11	ralbii 3077	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)
13		ralcom 3267	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)
14	12, 13	bitri 275	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)
15	14	anbi2i 623	. . . . 5 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
16	8, 15	bitri 275	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
17	1, 7, 16	3bitr4g 314	. . 3 ⊢ (𝐵 ≠ ∅ → (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ 𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶))
18	17	eqrdv 2728	. 2 ⊢ (𝐵 ≠ ∅ → ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶)
19	18	eqcomd 2736	1 ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2927  ∀wral 3046  Vcvv 3455  ∅c0 4304  ∩ ciin 4964   Fn wfn 6514  ‘cfv 6519  Xcixp 8874

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-12 2178  ax-ext 2702  ax-nul 5269

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2928  df-ral 3047  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iin 4966  df-br 5116  df-opab 5178  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-iota 6472  df-fun 6521  df-fn 6522  df-fv 6527  df-ixp 8875

This theorem is referenced by:  ixpint  8902  ptbasfi  23474  iccvonmbllem  46649