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Theorem sscres 16605
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 3941 . . 3 (𝑆𝑇) ⊆ 𝑆
2 inss2 3942 . . . . . . 7 (𝑆𝑇) ⊆ 𝑇
3 simpl 474 . . . . . . 7 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑥 ∈ (𝑆𝑇))
42, 3sseldi 3707 . . . . . 6 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑥𝑇)
5 simpr 479 . . . . . . 7 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑦 ∈ (𝑆𝑇))
62, 5sseldi 3707 . . . . . 6 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → 𝑦𝑇)
74, 6ovresd 6918 . . . . 5 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦))
8 eqimss 3763 . . . . 5 ((𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) = (𝑥𝐻𝑦) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
97, 8syl 17 . . . 4 ((𝑥 ∈ (𝑆𝑇) ∧ 𝑦 ∈ (𝑆𝑇)) → (𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
109rgen2a 3079 . . 3 𝑥 ∈ (𝑆𝑇)∀𝑦 ∈ (𝑆𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦)
111, 10pm3.2i 470 . 2 ((𝑆𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆𝑇)∀𝑦 ∈ (𝑆𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))
12 simpl 474 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝐻 Fn (𝑆 × 𝑆))
13 inss1 3941 . . . . 5 ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)
14 fnssres 6117 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ⊆ (𝑆 × 𝑆)) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
1512, 13, 14sylancl 697 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
16 resres 5519 . . . . . 6 ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)))
17 fnresdm 6113 . . . . . . . 8 (𝐻 Fn (𝑆 × 𝑆) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
1817adantr 472 . . . . . . 7 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑆 × 𝑆)) = 𝐻)
1918reseq1d 5502 . . . . . 6 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝐻 ↾ (𝑆 × 𝑆)) ↾ (𝑇 × 𝑇)) = (𝐻 ↾ (𝑇 × 𝑇)))
2016, 19syl5eqr 2772 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) = (𝐻 ↾ (𝑇 × 𝑇)))
21 inxp 5362 . . . . . 6 ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆𝑇) × (𝑆𝑇))
2221a1i 11 . . . . 5 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) = ((𝑆𝑇) × (𝑆𝑇)))
2320, 22fneq12d 6096 . . . 4 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝐻 ↾ ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇))) Fn ((𝑆 × 𝑆) ∩ (𝑇 × 𝑇)) ↔ (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆𝑇) × (𝑆𝑇))))
2415, 23mpbid 222 . . 3 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) Fn ((𝑆𝑇) × (𝑆𝑇)))
25 simpr 479 . . 3 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → 𝑆𝑉)
2624, 12, 25isssc 16602 . 2 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → ((𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻 ↔ ((𝑆𝑇) ⊆ 𝑆 ∧ ∀𝑥 ∈ (𝑆𝑇)∀𝑦 ∈ (𝑆𝑇)(𝑥(𝐻 ↾ (𝑇 × 𝑇))𝑦) ⊆ (𝑥𝐻𝑦))))
2711, 26mpbiri 248 1 ((𝐻 Fn (𝑆 × 𝑆) ∧ 𝑆𝑉) → (𝐻 ↾ (𝑇 × 𝑇)) ⊆cat 𝐻)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1596  wcel 2103  wral 3014  cin 3679  wss 3680   class class class wbr 4760   × cxp 5216  cres 5220   Fn wfn 5996  (class class class)co 6765  cat cssc 16589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-ov 6768  df-ixp 8026  df-ssc 16592
This theorem is referenced by:  sscid  16606  fullsubc  16632
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