Theorem mptelixpg 6851

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 2791	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
2		nfcv 2352	. . . . . 6 ⊢ Ⅎ𝑦𝐾
3		nfcsb1v 3137	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐾
4		csbeq1a 3113	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐾 = ⦋𝑦 / 𝑥⦌𝐾)
5	2, 3, 4	cbvixp 6832	. . . . 5 ⊢ X𝑥 ∈ 𝐼 𝐾 = X𝑦 ∈ 𝐼 ⦋𝑦 / 𝑥⦌𝐾
6	5	eleq2i 2276	. . . 4 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑦 ∈ 𝐼 ⦋𝑦 / 𝑥⦌𝐾)
7		elixp2 6819	. . . 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 987	. . . 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 2209	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 ↦ 𝐽) = (𝑥 ∈ 𝐼 ↦ 𝐽)
11	10	fnmpt 5426	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 → (𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼)
12	10	fvmpt2 5691	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = 𝐽)
13		simpr 110	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → 𝐽 ∈ 𝐾)
14	12, 13	eqeltrd 2286	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾)
15	14	ralimiaa 2572	. . . . . . 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 5451	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ↔ (𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶V)
18	10	fmpt 5758	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ V ↔ (𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶V)
19	10	fvmpt2 5691	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ V) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = 𝐽)
20	19	eleq1d 2278	. . . . . . . . . . . 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 2572	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ V → ∀𝑥 ∈ 𝐼 (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → 𝐽 ∈ 𝐾))
23		ralim 2569	. . . . . . . . . 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 1554	. . . . . . 7 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾
30		nffvmpt1 5614	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦)
31	30, 3	nfel 2361	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾
32		fveq2 5603	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦))
33	32, 4	eleq12d 2280	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾))
34	29, 31, 33	cbvral 2741	. . . . . 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 5837	. . . . 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 983   ∈ wcel 2180  ∀wral 2488  Vcvv 2779  ⦋csb 3104   ↦ cmpt 4124   Fn wfn 5289  ⟶wf 5290  ‘cfv 5294  Xcixp 6815

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ixp 6816

This theorem is referenced by:  prdsbasmpt  13279  prdsbasmpt2  13287