Theorem mptelixpg 8993

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 3509	. 2 ⊢ (𝐼 ∈ 𝑉 → 𝐼 ∈ V)
2		nfcv 2908	. . . . . 6 ⊢ Ⅎ𝑦𝐾
3		nfcsb1v 3946	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐾
4		csbeq1a 3935	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐾 = ⦋𝑦 / 𝑥⦌𝐾)
5	2, 3, 4	cbvixp 8972	. . . . 5 ⊢ X𝑥 ∈ 𝐼 𝐾 = X𝑦 ∈ 𝐼 ⦋𝑦 / 𝑥⦌𝐾
6	5	eleq2i 2836	. . . 4 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑥 ∈ 𝐼 𝐾 ↔ (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ X𝑦 ∈ 𝐼 ⦋𝑦 / 𝑥⦌𝐾)
7		elixp2 8959	. . . 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 1095	. . . 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 297	. . 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 2740	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐼 ↦ 𝐽) = (𝑥 ∈ 𝐼 ↦ 𝐽)
11	10	fnmpt 6720	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 → (𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼)
12	10	fvmpt2 7040	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = 𝐽)
13		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → 𝐽 ∈ 𝐾)
14	12, 13	eqeltrd 2844	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ 𝐾) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾)
15	14	ralimiaa 3088	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 → ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾)
16	11, 15	jca 511	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 → ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾))
17		dffn2 6749	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ↔ (𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶V)
18	10	fmpt 7144	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ V ↔ (𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶V)
19	10	fvmpt2 7040	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ V) → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = 𝐽)
20	19	eleq1d 2829	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ V) → (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 ↔ 𝐽 ∈ 𝐾))
21	20	biimpd 229	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐼 ∧ 𝐽 ∈ V) → (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → 𝐽 ∈ 𝐾))
22	21	ralimiaa 3088	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ V → ∀𝑥 ∈ 𝐼 (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → 𝐽 ∈ 𝐾))
23		ralim 3092	. . . . . . . . . 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 235	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽):𝐼⟶V → (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
26	17, 25	sylbi 217	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 → (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾))
27	26	imp 406	. . . . . 6 ⊢ (((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾) → ∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾)
28	16, 27	impbii 209	. . . . 5 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾))
29		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾
30		nffvmpt1 6931	. . . . . . . 8 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦)
31	30, 3	nfel 2923	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾
32		fveq2 6920	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) = ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦))
33	32, 4	eleq12d 2838	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾))
34	29, 31, 33	cbvralw 3312	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑥) ∈ 𝐾 ↔ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾)
35	34	anbi2i 622	. . . . 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 275	. . . 4 ⊢ (∀𝑥 ∈ 𝐼 𝐽 ∈ 𝐾 ↔ ((𝑥 ∈ 𝐼 ↦ 𝐽) Fn 𝐼 ∧ ∀𝑦 ∈ 𝐼 ((𝑥 ∈ 𝐼 ↦ 𝐽)‘𝑦) ∈ ⦋𝑦 / 𝑥⦌𝐾))
37		mptexg 7258	. . . . 5 ⊢ (𝐼 ∈ V → (𝑥 ∈ 𝐼 ↦ 𝐽) ∈ V)
38	37	biantrurd 532	. . . 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 284	. . 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 283	. 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 206   ∧ wa 395   ∧ w3a 1087   ∈ wcel 2108  ∀wral 3067  Vcvv 3488  ⦋csb 3921   ↦ cmpt 5249   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  Xcixp 8955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ixp 8956

This theorem is referenced by:  resixpfo  8994  ixpiunwdom  9659  dfac9  10206  prdsbasmpt  17530  prdsbasmpt2  17542  idfucl  17945  fuccocl  18034  fucidcl  18035  invfuc  18044  curf2cl  18301  yonedalem4c  18347  dprdwd  20055  ptpjopn  23641  dfac14lem  23646  ptcnplem  23650  ptcnp  23651  ptcn  23656  ptcmplem2  24082  tmdgsum2  24125  upixp  37689  kelac1  43020