Theorem sscres 16922

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 4125	. . 3 ⊢ (𝑆 ∩ 𝑇) ⊆ 𝑆
2		simpl 483	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑥 ∈ (𝑆 ∩ 𝑇))
3	2	elin2d 4097	. . . . . 6 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑥 ∈ 𝑇)
4		simpr 485	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑦 ∈ (𝑆 ∩ 𝑇))
5	4	elin2d 4097	. . . . . 6 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑦 ∈ 𝑇)
6	3, 5	ovresd 7171	. . . . 5 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦))
7		eqimss 3944	. . . . 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	rgen2a 3193	. . 3 ⊢ ∀𝑥 ∈ (𝑆 ∩ 𝑇)∀𝑦 ∈ (𝑆 ∩ 𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦)
10	1, 9	pm3.2i 471	. 2 ⊢ ((𝑆 ∩ 𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆 ∩ 𝑇)∀𝑦 ∈ (𝑆 ∩ 𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
11		simpl 483	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → 𝐻 Fn (𝑆 × 𝑆))
12		inss1 4125	. . . . 5 ⊢ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)
13		fnssres 6340	. . . . 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 5747	. . . . . 6 ⊢ ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
16		fnresdm 6336	. . . . . . . 8 ⊢ (𝐻 Fn (𝑆 × 𝑆) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
17	16	adantr 481	. . . . . . 7 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
18	17	reseq1d 5733	. . . . . 6 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ (𝑇 × 𝑇)))
19	15, 18	syl5eqr 2845	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) = (𝐻 ↾ (𝑇 × 𝑇)))
20		inxp 5589	. . . . . 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 6318	. . . 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 233	. . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇)))
24		simpr 485	. . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → 𝑆 ∈ 𝑉)
25	23, 11, 24	isssc 16919	. 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 259	1 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1522   ∈ wcel 2081  ∀wral 3105   ∩ cin 3858   ⊆ wss 3859   class class class wbr 4962   × cxp 5441   ↾ cres 5445   Fn wfn 6220  (class class class)co 7016   ⊆_cat cssc 16906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-ixp 8311  df-ssc 16909

This theorem is referenced by:  sscid  16923  fullsubc  16949