Theorem ixpin 6891

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 594	. . . 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 3390	. . . . . . 7 ⊢ ((𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶))
3	2	ralbii 2538	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ ∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶))
4		r19.26 2659	. . . . . 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 2805	. . . . . 6 ⊢ 𝑓 ∈ V
8	7	elixp 6873	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
9	7	elixp 6873	. . . . 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 6873	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶)))
13		elin 3390	. . 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 2228	1 ⊢ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ∀wral 2510   ∩ cin 3199   Fn wfn 5321  ‘cfv 5326  Xcixp 6866

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ixp 6867

This theorem is referenced by: (None)