Theorem ixpin 8930

Description: The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)

Assertion

Ref	Expression
ixpin	⊢ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem ixpin

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anandi 689	. . . 4 ⊢ ((𝑓 Fn 𝐴 ∧ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)))
2		elin 3915	. . . . . . 7 ⊢ ((𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶))
3	2	ralbii 3108	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ ∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶))
4		r19.26 3122	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
5	3, 4	bitri 278	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
6	5	anbi2i 635	. . . 4 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶)) ↔ (𝑓 Fn 𝐴 ∧ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)))
7		vex 3454	. . . . . 6 ⊢ 𝑓 ∈ V
8	7	elixp 8911	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
9	7	elixp 8911	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
10	8, 9	anbi12i 640	. . . 4 ⊢ ((𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)))
11	1, 6, 10	3bitr4i 306	. . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶)) ↔ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶))
12	7	elixp 8911	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶)))
13		elin 3915	. . 3 ⊢ (𝑓 ∈ (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶) ↔ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶))
14	11, 12, 13	3bitr4i 306	. 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ 𝑓 ∈ (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶))
15	14	eqriv 2757	1 ⊢ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∩ cin 3898   Fn wfn 6528  ‘cfv 6533  Xcixp 8904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541  df-ixp 8905

This theorem is used by:  ptbasin  23803  ptclsg  23841  ptrest  38368