Theorem inixp 35018

Description: Intersection of Cartesian products over the same base set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem inixp

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		an4 654	. . . 4 ⊢ (((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)) ↔ ((𝑓 Fn 𝐴 ∧ 𝑓 Fn 𝐴) ∧ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)))
2		anidm 567	. . . . 5 ⊢ ((𝑓 Fn 𝐴 ∧ 𝑓 Fn 𝐴) ↔ 𝑓 Fn 𝐴)
3		r19.26 3170	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶) ↔ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
4		elin 4169	. . . . . . . 8 ⊢ ((𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶) ↔ ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶))
5	4	bicomi 226	. . . . . . 7 ⊢ (((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶))
6	5	ralbii 3165	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 ((𝑓‘𝑥) ∈ 𝐵 ∧ (𝑓‘𝑥) ∈ 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶))
7	3, 6	bitr3i 279	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶))
8	2, 7	anbi12i 628	. . . 4 ⊢ (((𝑓 Fn 𝐴 ∧ 𝑓 Fn 𝐴) ∧ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶)))
9	1, 8	bitri 277	. . 3 ⊢ (((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶)))
10		vex 3497	. . . . 5 ⊢ 𝑓 ∈ V
11	10	elixp 8468	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵))
12	10	elixp 8468	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
13	11, 12	anbi12i 628	. . 3 ⊢ ((𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶) ↔ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) ∧ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)))
14	10	elixp 8468	. . 3 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ (𝐵 ∩ 𝐶)))
15	9, 13, 14	3bitr4i 305	. 2 ⊢ ((𝑓 ∈ X𝑥 ∈ 𝐴 𝐵 ∧ 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶) ↔ 𝑓 ∈ X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶))
16	15	ineqri 4180	1 ⊢ (X𝑥 ∈ 𝐴 𝐵 ∩ X𝑥 ∈ 𝐴 𝐶) = X𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)

Syntax hints:   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138   ∩ cin 3935   Fn wfn 6350  ‘cfv 6355  Xcixp 8461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-iota 6314  df-fun 6357  df-fn 6358  df-fv 6363  df-ixp 8462

This theorem is referenced by: (None)