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Theorem sscres 17083
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) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)

Proof of Theorem sscres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 4204 . . 3 (𝑆𝑇) ⊆ 𝑆
2 simpl 483 . . . . . . 7 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑥 ∈ (𝑆𝑇))
32elin2d 4175 . . . . . 6 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑥𝑇)
4 simpr 485 . . . . . . 7 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑦 ∈ (𝑆𝑇))
54elin2d 4175 . . . . . 6 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑦𝑇)
63, 5ovresd 7304 . . . . 5 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦))
7 eqimss 4022 . . . . 5 ((𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
86, 7syl 17 . . . 4 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
98rgen2 3203 . . 3 𝑥 ∈ (𝑆𝑇)∀𝑦 ∈ (𝑆𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦)
101, 9pm3.2i 471 . 2 ((𝑆𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆𝑇)∀𝑦 ∈ (𝑆𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
11 simpl 483 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝐻 Fn (𝑆 × 𝑆))
12 inss1 4204 . . . . 5 ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)
13 fnssres 6464 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
1411, 12, 13sylancl 586 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
15 resres 5860 . . . . . 6 ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
16 fnresdm 6460 . . . . . . . 8 (𝐻 Fn (𝑆 × 𝑆) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
1716adantr 481 . . . . . . 7 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
1817reseq1d 5846 . . . . . 6 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ (𝑇 × 𝑇)))
1915, 18syl5eqr 2870 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) = (𝐻 ↾ (𝑇 × 𝑇)))
20 inxp 5697 . . . . . 6 ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆𝑇) × (𝑆𝑇))
2120a1i 11 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆𝑇) × (𝑆𝑇)))
2219, 21fneq12d 6442 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ↔ (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆𝑇) × (𝑆𝑇))))
2314, 22mpbid 233 . . 3 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆𝑇) × (𝑆𝑇)))
24 simpr 485 . . 3 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝑆𝑉)
2523, 11, 24isssc 17080 . 2 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻 ↔ ((𝑆𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆𝑇)∀𝑦 ∈ (𝑆𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))))
2610, 25mpbiri 259 1 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3138  cin 3934  wss 3935   class class class wbr 5058   × cxp 5547  cres 5551   Fn wfn 6344  (class class class)co 7145  cat cssc 17067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-ixp 8451  df-ssc 17070
This theorem is referenced by:  sscid  17084  fullsubc  17110
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