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Theorem sscres 17884
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 4258 . . 3 (𝑆𝑇) ⊆ 𝑆
2 simpl 482 . . . . . . 7 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑥 ∈ (𝑆𝑇))
32elin2d 4228 . . . . . 6 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑥𝑇)
4 simpr 484 . . . . . . 7 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑦 ∈ (𝑆𝑇))
54elin2d 4228 . . . . . 6 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑦𝑇)
63, 5ovresd 7617 . . . . 5 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦))
7 eqimss 4067 . . . . 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 3205 . . 3 𝑥 ∈ (𝑆𝑇)∀𝑦 ∈ (𝑆𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦)
101, 9pm3.2i 470 . 2 ((𝑆𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆𝑇)∀𝑦 ∈ (𝑆𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
11 simpl 482 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝐻 Fn (𝑆 × 𝑆))
12 inss1 4258 . . . . 5 ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)
13 fnssres 6703 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
1411, 12, 13sylancl 585 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
15 resres 6022 . . . . . 6 ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
16 fnresdm 6699 . . . . . . . 8 (𝐻 Fn (𝑆 × 𝑆) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
1716adantr 480 . . . . . . 7 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
1817reseq1d 6008 . . . . . 6 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ (𝑇 × 𝑇)))
1915, 18eqtr3id 2794 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) = (𝐻 ↾ (𝑇 × 𝑇)))
20 inxp 5856 . . . . . 6 ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆𝑇) × (𝑆𝑇))
2120a1i 11 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆𝑇) × (𝑆𝑇)))
2219, 21fneq12d 6674 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ↔ (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆𝑇) × (𝑆𝑇))))
2314, 22mpbid 232 . . 3 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆𝑇) × (𝑆𝑇)))
24 simpr 484 . . 3 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝑆𝑉)
2523, 11, 24isssc 17881 . 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 258 1 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  wral 3067  cin 3975  wss 3976   class class class wbr 5166   × cxp 5698  cres 5702   Fn wfn 6568  (class class class)co 7448  cat cssc 17868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-ixp 8956  df-ssc 17871
This theorem is referenced by:  sscid  17885  fullsubc  17914
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