Theorem ixpiinm 6783

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 2257	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵))
2	1	cbvexv 1933	. . 3 ⊢ (∃𝑦 𝑦 ∈ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵)
3		r19.28mv 3543	. . . . 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 3921	. . . . . . 7 ⊢ (𝑓 ∈ V → (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶))
5	4	elv 2767	. . . . . 6 ⊢ (𝑓 ∈ ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑓 ∈ X𝑥 ∈ 𝐴 𝐶)
6		vex 2766	. . . . . . . 8 ⊢ 𝑓 ∈ V
7	6	elixp 6764	. . . . . . 7 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
8	7	ralbii 2503	. . . . . 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 6764	. . . . . 6 ⊢ (𝑓 ∈ X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ (𝑓 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶))
11		vex 2766	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
12	6, 11	fvex 5578	. . . . . . . . . 10 ⊢ (𝑓‘𝑥) ∈ V
13		eliin 3921	. . . . . . . . . 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 2503	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝑓‘𝑥) ∈ 𝐶)
16		ralcom 2660	. . . . . . . 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 2194	. . 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 2202	1 ⊢ (∃𝑧 𝑧 ∈ 𝐵 → X𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 = ∩ 𝑦 ∈ 𝐵 X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  ∩ ciin 3917   Fn wfn 5253  ‘cfv 5258  Xcixp 6757

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iin 3919  df-br 4034  df-opab 4095  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ixp 6758

This theorem is referenced by:  ixpintm  6784