Theorem ixpin 6750

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 590	. . . 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 3333	. . . . . . 7 ⊢ ((𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶))
3	2	ralbii 2496	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ ∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶))
4		r19.26 2616	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
5	3, 4	bitri 184	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
6	5	anbi2i 457	. . . 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 2755	. . . . . 6 ⊢ 𝑓 ∈ V
8	7	elixp 6732	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
9	7	elixp 6732	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
10	8, 9	anbi12i 460	. . . 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 212	. . 3 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶)) ↔ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶))
12	7	elixp 6732	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶)))
13		elin 3333	. . 3 ⊢ (𝑓 ∈ (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶) ↔ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶))
14	11, 12, 13	3bitr4i 212	. 2 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ 𝑓 ∈ (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶))
15	14	eqriv 2186	1 ⊢ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2160  ∀wral 2468   ∩ cin 3143   Fn wfn 5230  ‘cfv 5235  Xcixp 6725

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ixp 6726

This theorem is referenced by: (None)