Theorem sscres 17779

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 4167	. . 3 ⊢ (𝑆 ∩ 𝑇) ⊆ 𝑆
2		simpl 482	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑥 ∈ (𝑆 ∩ 𝑇))
3	2	elin2d 4136	. . . . . 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 4136	. . . . . 6 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑦 ∈ 𝑇)
6	3, 5	ovresd 7523	. . . . 5 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦))
7		eqimss 3975	. . . . 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 3175	. . 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 4167	. . . . 5 ⊢ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)
13		fnssres 6610	. . . . 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 587	. . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
15		resres 5946	. . . . . 6 ⊢ ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
16		fnresdm 6606	. . . . . . . 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 5932	. . . . . 6 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ (𝑇 × 𝑇)))
19	15, 18	eqtr3id 2784	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) = (𝐻 ↾ (𝑇 × 𝑇)))
20		inxp 5776	. . . . . 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 6582	. . . 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 17776	. 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 1542   ∈ wcel 2114  ∀wral 3049   ∩ cin 3884   ⊆ wss 3885   class class class wbr 5074   × cxp 5618   ↾ cres 5622   Fn wfn 6482  (class class class)co 7356   ⊆_cat cssc 17763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-ixp 8835  df-ssc 17766

This theorem is referenced by:  sscid  17780  fullsubc  17806