Theorem ixpin 8873

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 677	. . . 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 3919	. . . . . . 7 ⊢ ((𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶))
3	2	ralbii 3084	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ ∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶))
4		r19.26 3098	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
5	3, 4	bitri 275	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
6	5	anbi2i 624	. . . 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 3446	. . . . . 6 ⊢ 𝑓 ∈ V
8	7	elixp 8854	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
9	7	elixp 8854	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
10	8, 9	anbi12i 629	. . . 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 303	. . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶)) ↔ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶))
12	7	elixp 8854	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶)))
13		elin 3919	. . 3 ⊢ (𝑓 ∈ (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶) ↔ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶))
14	11, 12, 13	3bitr4i 303	. 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ 𝑓 ∈ (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶))
15	14	eqriv 2734	1 ⊢ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∩ cin 3902   Fn wfn 6495  ‘cfv 6500  Xcixp 8847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508  df-ixp 8848

This theorem is referenced by:  ptbasin  23533  ptclsg  23571  ptrest  37864