Theorem sscres 17785

Description: Any function restricted to a square domain is a subcategory subset of the original. (Contributed by Mario Carneiro, 6-Jan-2017.)

Assertion

Ref	Expression
sscres	⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)

Proof of Theorem sscres

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 4200	. . 3 ⊢ (𝑆 ∩ 𝑇) ⊆ 𝑆
2		simpl 482	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑥 ∈ (𝑆 ∩ 𝑇))
3	2	elin2d 4168	. . . . . 6 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑥 ∈ 𝑇)
4		simpr 484	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑦 ∈ (𝑆 ∩ 𝑇))
5	4	elin2d 4168	. . . . . 6 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑦 ∈ 𝑇)
6	3, 5	ovresd 7556	. . . . 5 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦))
7		eqimss 4005	. . . . 5 ⊢ ((𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
8	6, 7	syl 17	. . . 4 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
9	8	rgen2 3177	. . 3 ⊢ ∀𝑥 ∈ (𝑆 ∩ 𝑇)∀𝑦 ∈ (𝑆 ∩ 𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦)
10	1, 9	pm3.2i 470	. 2 ⊢ ((𝑆 ∩ 𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆 ∩ 𝑇)∀𝑦 ∈ (𝑆 ∩ 𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
11		simpl 482	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → 𝐻 Fn (𝑆 × 𝑆))
12		inss1 4200	. . . . 5 ⊢ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)
13		fnssres 6641	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
14	11, 12, 13	sylancl 586	. . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
15		resres 5963	. . . . . 6 ⊢ ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
16		fnresdm 6637	. . . . . . . 8 ⊢ (𝐻 Fn (𝑆 × 𝑆) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
17	16	adantr 480	. . . . . . 7 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
18	17	reseq1d 5949	. . . . . 6 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ (𝑇 × 𝑇)))
19	15, 18	eqtr3id 2778	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) = (𝐻 ↾ (𝑇 × 𝑇)))
20		inxp 5795	. . . . . 6 ⊢ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇))
21	20	a1i 11	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇)))
22	19, 21	fneq12d 6613	. . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ↔ (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇))))
23	14, 22	mpbid 232	. . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇)))
24		simpr 484	. . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → 𝑆 ∈ 𝑉)
25	23, 11, 24	isssc 17782	. 2 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻 ↔ ((𝑆 ∩ 𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆 ∩ 𝑇)∀𝑦 ∈ (𝑆 ∩ 𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))))
26	10, 25	mpbiri 258	1 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ∩ cin 3913   ⊆ wss 3914   class class class wbr 5107   × cxp 5636   ↾ cres 5640   Fn wfn 6506  (class class class)co 7387   ⊆_cat cssc 17769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-ixp 8871  df-ssc 17772

This theorem is referenced by:  sscid  17786  fullsubc  17812