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Theorem sscres 17714
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 4192 . . 3 (𝑆𝑇) ⊆ 𝑆
2 simpl 484 . . . . . . 7 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑥 ∈ (𝑆𝑇))
32elin2d 4163 . . . . . 6 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑥𝑇)
4 simpr 486 . . . . . . 7 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑦 ∈ (𝑆𝑇))
54elin2d 4163 . . . . . 6 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑦𝑇)
63, 5ovresd 7525 . . . . 5 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦))
7 eqimss 4004 . . . . 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 3191 . . 3 𝑥 ∈ (𝑆𝑇)∀𝑦 ∈ (𝑆𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦)
101, 9pm3.2i 472 . 2 ((𝑆𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆𝑇)∀𝑦 ∈ (𝑆𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
11 simpl 484 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝐻 Fn (𝑆 × 𝑆))
12 inss1 4192 . . . . 5 ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)
13 fnssres 6628 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
1411, 12, 13sylancl 587 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
15 resres 5954 . . . . . 6 ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
16 fnresdm 6624 . . . . . . . 8 (𝐻 Fn (𝑆 × 𝑆) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
1716adantr 482 . . . . . . 7 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
1817reseq1d 5940 . . . . . 6 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ (𝑇 × 𝑇)))
1915, 18eqtr3id 2787 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) = (𝐻 ↾ (𝑇 × 𝑇)))
20 inxp 5792 . . . . . 6 ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆𝑇) × (𝑆𝑇))
2120a1i 11 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆𝑇) × (𝑆𝑇)))
2219, 21fneq12d 6601 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ↔ (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆𝑇) × (𝑆𝑇))))
2314, 22mpbid 231 . . 3 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆𝑇) × (𝑆𝑇)))
24 simpr 486 . . 3 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝑆𝑉)
2523, 11, 24isssc 17711 . 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 397   = wceq 1542  wcel 2107  wral 3061  cin 3913  wss 3914   class class class wbr 5109   × cxp 5635  cres 5639   Fn wfn 6495  (class class class)co 7361  cat cssc 17698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-ixp 8842  df-ssc 17701
This theorem is referenced by:  sscid  17715  fullsubc  17744
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