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Theorem ssc2 17751
Description: Infer subset relation on morphisms from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
ssc2.1 (𝜑𝐻 Fn (𝑆 × 𝑆))
ssc2.2 (𝜑𝐻cat 𝐽)
ssc2.3 (𝜑𝑋𝑆)
ssc2.4 (𝜑𝑌𝑆)
Assertion
Ref Expression
ssc2 (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))

Proof of Theorem ssc2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssc2.3 . 2 (𝜑𝑋𝑆)
2 ssc2.4 . 2 (𝜑𝑌𝑆)
3 ssc2.2 . . . 4 (𝜑𝐻cat 𝐽)
4 ssc2.1 . . . . 5 (𝜑𝐻 Fn (𝑆 × 𝑆))
5 eqidd 2732 . . . . . 6 (𝜑 → dom dom 𝐽 = dom dom 𝐽)
63, 5sscfn2 17747 . . . . 5 (𝜑𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
7 sscrel 17742 . . . . . . 7 Rel ⊆cat
87brrelex2i 5725 . . . . . 6 (𝐻cat 𝐽𝐽 ∈ V)
9 dmexg 7876 . . . . . 6 (𝐽 ∈ V → dom 𝐽 ∈ V)
10 dmexg 7876 . . . . . 6 (dom 𝐽 ∈ V → dom dom 𝐽 ∈ V)
113, 8, 9, 104syl 19 . . . . 5 (𝜑 → dom dom 𝐽 ∈ V)
124, 6, 11isssc 17749 . . . 4 (𝜑 → (𝐻cat 𝐽 ↔ (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
133, 12mpbid 231 . . 3 (𝜑 → (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
1413simprd 496 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
15 oveq1 7400 . . . 4 (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16 oveq1 7400 . . . 4 (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦))
1715, 16sseq12d 4011 . . 3 (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦)))
18 oveq2 7401 . . . 4 (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
19 oveq2 7401 . . . 4 (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌))
2018, 19sseq12d 4011 . . 3 (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)))
2117, 20rspc2va 3619 . 2 (((𝑋𝑆𝑌𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
221, 2, 14, 21syl21anc 836 1 (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wral 3060  Vcvv 3473  wss 3944   class class class wbr 5141   × cxp 5667  dom cdm 5669   Fn wfn 6527  (class class class)co 7393  cat cssc 17736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-ixp 8875  df-ssc 17739
This theorem is referenced by:  ssctr  17754  ssceq  17755  subcss2  17775
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