Theorem mptelixpg 6903

Description: Condition for an explicit member of an indexed product. (Contributed by Stefan O'Rear, 4-Jan-2015.)

Assertion

Ref	Expression
mptelixpg	⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))

Distinct variable group:   𝑥,𝐼

Allowed substitution hints:   𝐽(𝑥)   𝐾(𝑥)   𝑉(𝑥)

Proof of Theorem mptelixpg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2814	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
2		nfcv 2374	. . . . . 6 ⊢ Ⅎ𝑦𝐾
3		nfcsb1v 3160	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐾
4		csbeq1a 3136	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐾 = ⦋𝑦 / 𝑥⦌𝐾)
5	2, 3, 4	cbvixp 6884	. . . . 5 ⊢ X𝑥 ∈ 𝐼 𝐾 = X𝑦 ∈ 𝐼 ⦋𝑦 / 𝑥⦌𝐾
6	5	eleq2i 2298	. . . 4 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑦 ∈ 𝐼 ⦋𝑦 / 𝑥⦌𝐾)
7		elixp2 6871	. . . 4 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑦 ∈ 𝐼 ⦋𝑦 / 𝑥⦌𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V ∧ (𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾))
8		3anass 1008	. . . 4 ⊢ (((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V ∧ (𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾) ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V ∧ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾)))
9	6, 7, 8	3bitri 206	. . 3 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V ∧ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾)))
10		eqid 2231	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 ↦ 𝐽) = (𝑥 ∈ 𝐼 ↦ 𝐽)
11	10	fnmpt 5459	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 → (𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼)
12	10	fvmpt2 5730	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = 𝐽)
13		simpr 110	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → 𝐽 ∈ 𝐾)
14	12, 13	eqeltrd 2308	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾)
15	14	ralimiaa 2594	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 → ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾)
16	11, 15	jca 306	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 → ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾))
17		dffn2 5484	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ↔ (𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶V)
18	10	fmpt 5797	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ V ↔ (𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶V)
19	10	fvmpt2 5730	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ V) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = 𝐽)
20	19	eleq1d 2300	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ V) → (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 ↔ 𝐽 ∈ 𝐾))
21	20	biimpd 144	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ V) → (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → 𝐽 ∈ 𝐾))
22	21	ralimiaa 2594	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ V → ∀𝑥 ∈ 𝐼 (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → 𝐽 ∈ 𝐾))
23		ralim 2591	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐼 (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → 𝐽 ∈ 𝐾) → (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
24	22, 23	syl 14	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ V → (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
25	18, 24	sylbir 135	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶V → (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
26	17, 25	sylbi 121	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 → (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
27	26	imp 124	. . . . . 6 ⊢ (((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾) → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾)
28	16, 27	impbii 126	. . . . 5 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾))
29		nfv 1576	. . . . . . 7 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾
30		nffvmpt1 5650	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦)
31	30, 3	nfel 2383	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾
32		fveq2 5639	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦))
33	32, 4	eleq12d 2302	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾))
34	29, 31, 33	cbvral 2763	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 ↔ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾)
35	34	anbi2i 457	. . . . 5 ⊢ (((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾) ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾))
36	28, 35	bitri 184	. . . 4 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾))
37		mptexg 5879	. . . . 5 ⊢ (𝐼 ∈ V → (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V)
38	37	biantrurd 305	. . . 4 ⊢ (𝐼 ∈ V → (((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾) ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V ∧ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾))))
39	36, 38	bitr2id 193	. . 3 ⊢ (𝐼 ∈ V → (((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V ∧ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾)) ↔ ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
40	9, 39	bitrid 192	. 2 ⊢ (𝐼 ∈ V → ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
41	1, 40	syl 14	1 ⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1004   ∈ wcel 2202  ∀wral 2510  Vcvv 2802  ⦋csb 3127   ↦ cmpt 4150   Fn wfn 5321  ⟶wf 5322  ‘cfv 5326  Xcixp 6867

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-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ixp 6868

This theorem is referenced by:  prdsbasmpt  13365  prdsbasmpt2  13373