Theorem ixpiin 8874

Description: The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.)

Assertion

Ref	Expression
ixpiin	⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem ixpiin

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.28zv 4461	. . . 4 ⊢ (𝐵 ≠ ∅ → (∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)))
2		eliin 4953	. . . . . 6 ⊢ (𝑓 ∈ V → (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶))
3	2	elv 3447	. . . . 5 ⊢ (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶)
4		vex 3446	. . . . . . 7 ⊢ 𝑓 ∈ V
5	4	elixp 8854	. . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
6	5	ralbii 3084	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
7	3, 6	bitri 275	. . . 4 ⊢ (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
8	4	elixp 8854	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶))
9		fvex 6855	. . . . . . . . 9 ⊢ (𝑓‘𝑥) ∈ V
10		eliin 4953	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) ∈ V → ((𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶))
11	9, 10	ax-mp 5	. . . . . . . 8 ⊢ ((𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)
12	11	ralbii 3084	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)
13		ralcom 3266	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)
14	12, 13	bitri 275	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)
15	14	anbi2i 624	. . . . 5 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
16	8, 15	bitri 275	. . . 4 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
17	1, 7, 16	3bitr4g 314	. . 3 ⊢ (𝐵 ≠ ∅ → (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ 𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶))
18	17	eqrdv 2735	. 2 ⊢ (𝐵 ≠ ∅ → ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶)
19	18	eqcomd 2743	1 ⊢ (𝐵 ≠ ∅ → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  Vcvv 3442  ∅c0 4287  ∩ ciin 4949   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-11 2163  ax-12 2185  ax-ext 2709  ax-nul 5253

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-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iin 4951  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:  ixpint  8875  ptbasfi  23537  iccvonmbllem  47036