Theorem ixpiin 8207

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 4290	. . . 4 ⊢ (𝐵 ≠ ∅ → (∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)))
2		vex 3417	. . . . . 6 ⊢ 𝑓 ∈ V
3		eliin 4747	. . . . . 6 ⊢ (𝑓 ∈ V → (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶))
4	2, 3	ax-mp 5	. . . . 5 ⊢ (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶)
5	2	elixp 8188	. . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
6	5	ralbii 3189	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
7	4, 6	bitri 267	. . . 4 ⊢ (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
8	2	elixp 8188	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶))
9		fvex 6450	. . . . . . . . 9 ⊢ (𝑓‘𝑥) ∈ V
10		eliin 4747	. . . . . . . . 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 3189	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)
13		ralcom 3308	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)
14	12, 13	bitri 267	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)
15	14	anbi2i 616	. . . . 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 267	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
17	1, 7, 16	3bitr4g 306	. . 3 ⊢ (𝐵 ≠ ∅ → (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ 𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶))
18	17	eqrdv 2823	. 2 ⊢ (𝐵 ≠ ∅ → ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶)
19	18	eqcomd 2831	1 ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ∀wral 3117  Vcvv 3414  ∅c0 4146  ∩ ciin 4743   Fn wfn 6122  ‘cfv 6127  Xcixp 8181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-nul 5015

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iin 4745  df-br 4876  df-opab 4938  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fn 6130  df-fv 6135  df-ixp 8182

This theorem is referenced by:  ixpint  8208  ptbasfi  21762  iccvonmbllem  41680