Theorem sscres 16483

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 3833	. . 3 ⊢ (𝑆 ∩ 𝑇) ⊆ 𝑆
2		inss2 3834	. . . . . . 7 ⊢ (𝑆 ∩ 𝑇) ⊆ 𝑇
3		simpl 473	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑥 ∈ (𝑆 ∩ 𝑇))
4	2, 3	sseldi 3601	. . . . . 6 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑥 ∈ 𝑇)
5		simpr 477	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑦 ∈ (𝑆 ∩ 𝑇))
6	2, 5	sseldi 3601	. . . . . 6 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → 𝑦 ∈ 𝑇)
7	4, 6	ovresd 6801	. . . . 5 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦))
8		eqimss 3657	. . . . 5 ⊢ ((𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
9	7, 8	syl 17	. . . 4 ⊢ ((𝑥 ∈ (𝑆 ∩ 𝑇) ∧ 𝑦 ∈ (𝑆 ∩ 𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
10	9	rgen2a 2977	. . 3 ⊢ ∀𝑥 ∈ (𝑆 ∩ 𝑇)∀𝑦 ∈ (𝑆 ∩ 𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦)
11	1, 10	pm3.2i 471	. 2 ⊢ ((𝑆 ∩ 𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆 ∩ 𝑇)∀𝑦 ∈ (𝑆 ∩ 𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
12		simpl 473	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → 𝐻 Fn (𝑆 × 𝑆))
13		inss1 3833	. . . . 5 ⊢ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)
14		fnssres 6004	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
15	12, 13, 14	sylancl 694	. . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
16		resres 5409	. . . . . 6 ⊢ ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
17		fnresdm 6000	. . . . . . . 8 ⊢ (𝐻 Fn (𝑆 × 𝑆) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
18	17	adantr 481	. . . . . . 7 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
19	18	reseq1d 5395	. . . . . 6 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ (𝑇 × 𝑇)))
20	16, 19	syl5eqr 2670	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) = (𝐻 ↾ (𝑇 × 𝑇)))
21		inxp 5254	. . . . . 6 ⊢ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇))
22	21	a1i 11	. . . . 5 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇)))
23	20, 22	fneq12d 5983	. . . 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))))
24	15, 23	mpbid 222	. . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆 ∩ 𝑇) × (𝑆 ∩ 𝑇)))
25		simpr 477	. . 3 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → 𝑆 ∈ 𝑉)
26	24, 12, 25	isssc 16480	. 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))))
27	11, 26	mpbiri 248	1 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆 ∈ 𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∩ cin 3573   ⊆ wss 3574   class class class wbr 4653   × cxp 5112   ↾ cres 5116   Fn wfn 5883  (class class class)co 6650   ⊆_cat cssc 16467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-ixp 7909  df-ssc 16470

This theorem is referenced by:  sscid  16484  fullsubc  16510