Theorem ixpiinm 6811

Description: The indexed intersection of a collection of infinite Cartesian products. (Contributed by Mario Carneiro, 6-Feb-2015.) (Revised by Jim Kingdon, 15-Feb-2023.)

Assertion

Ref	Expression
ixpiinm	⊢ (∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶)

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

Allowed substitution hints:   𝐴(𝑧)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem ixpiinm

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2266	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
2	1	cbvexv 1942	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵)
3		r19.28mv 3553	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)))
4		eliin 3932	. . . . . . 7 ⊢ (𝑓 ∈ V → (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶))
5	4	elv 2776	. . . . . 6 ⊢ (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶)
6		vex 2775	. . . . . . . 8 ⊢ 𝑓 ∈ V
7	6	elixp 6792	. . . . . . 7 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
8	7	ralbii 2512	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
9	5, 8	bitri 184	. . . . 5 ⊢ (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
10	6	elixp 6792	. . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶))
11		vex 2775	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
12	6, 11	fvex 5596	. . . . . . . . . 10 ⊢ (𝑓‘𝑥) ∈ V
13		eliin 3932	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ V → ((𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶))
14	12, 13	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)
15	14	ralbii 2512	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)
16		ralcom 2669	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)
17	15, 16	bitri 184	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)
18	17	anbi2i 457	. . . . . 6 ⊢ ((𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶) ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
19	10, 18	bitri 184	. . . . 5 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐵 ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
20	3, 9, 19	3bitr4g 223	. . . 4 ⊢ (∃𝑦 𝑦 ∈ 𝐵 → (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ 𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶))
21	20	eqrdv 2203	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐵 → ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶)
22	2, 21	sylbir 135	. 2 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 = X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶)
23	22	eqcomd 2211	1 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373  ∃wex 1515   ∈ wcel 2176  ∀wral 2484  Vcvv 2772  ∩ ciin 3928   Fn wfn 5266  ‘cfv 5271  Xcixp 6785

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iin 3930  df-br 4045  df-opab 4106  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-ixp 6786

This theorem is referenced by:  ixpintm  6812