Theorem mptelixpg 8952

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 3482	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
2		nfcv 2892	. . . . . 6 ⊢ Ⅎ𝑦𝐾
3		nfcsb1v 3909	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐾
4		csbeq1a 3898	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐾 = ⦋𝑦 / 𝑥⦌𝐾)
5	2, 3, 4	cbvixp 8931	. . . . 5 ⊢ X𝑥 ∈ 𝐼 𝐾 = X𝑦 ∈ 𝐼 ⦋𝑦 / 𝑥⦌𝐾
6	5	eleq2i 2817	. . . 4 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑦 ∈ 𝐼 ⦋𝑦 / 𝑥⦌𝐾)
7		elixp2 8918	. . . 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 1092	. . . 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 296	. . 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 2725	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 ↦ 𝐽) = (𝑥 ∈ 𝐼 ↦ 𝐽)
11	10	fnmpt 6690	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 → (𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼)
12	10	fvmpt2 7011	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = 𝐽)
13		simpr 483	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → 𝐽 ∈ 𝐾)
14	12, 13	eqeltrd 2825	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾)
15	14	ralimiaa 3072	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 → ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾)
16	11, 15	jca 510	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 → ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾))
17		dffn2 6719	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ↔ (𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶V)
18	10	fmpt 7115	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ V ↔ (𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶V)
19	10	fvmpt2 7011	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ V) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = 𝐽)
20	19	eleq1d 2810	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ V) → (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 ↔ 𝐽 ∈ 𝐾))
21	20	biimpd 228	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ V) → (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → 𝐽 ∈ 𝐾))
22	21	ralimiaa 3072	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ V → ∀𝑥 ∈ 𝐼 (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → 𝐽 ∈ 𝐾))
23		ralim 3076	. . . . . . . . . 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 17	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ V → (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
25	18, 24	sylbir 234	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶V → (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
26	17, 25	sylbi 216	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 → (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
27	26	imp 405	. . . . . 6 ⊢ (((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾) → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾)
28	16, 27	impbii 208	. . . . 5 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾))
29		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾
30		nffvmpt1 6903	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦)
31	30, 3	nfel 2907	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾
32		fveq2 6892	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦))
33	32, 4	eleq12d 2819	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾))
34	29, 31, 33	cbvralw 3294	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 ↔ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾)
35	34	anbi2i 621	. . . . 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 274	. . . 4 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾))
37		mptexg 7229	. . . . 5 ⊢ (𝐼 ∈ V → (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V)
38	37	biantrurd 531	. . . 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 283	. . 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 282	. 2 ⊢ (𝐼 ∈ V → ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
41	1, 40	syl 17	1 ⊢ (𝐼 ∈ 𝑉 → ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   ∈ wcel 2098  ∀wral 3051  Vcvv 3463  ⦋csb 3884   ↦ cmpt 5226   Fn wfn 6538  ⟶wf 6539  ‘cfv 6543  Xcixp 8914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ixp 8915

This theorem is referenced by:  resixpfo  8953  ixpiunwdom  9613  dfac9  10159  prdsbasmpt  17451  prdsbasmpt2  17463  idfucl  17866  fuccocl  17955  fucidcl  17956  invfuc  17965  curf2cl  18222  yonedalem4c  18268  dprdwd  19972  ptpjopn  23534  dfac14lem  23539  ptcnplem  23543  ptcnp  23544  ptcn  23549  ptcmplem2  23975  tmdgsum2  24018  upixp  37259  kelac1  42552