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Theorem ssc2 16406
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 2622 . . . . . 6 (𝜑 → dom dom 𝐽 = dom dom 𝐽)
63, 5sscfn2 16402 . . . . 5 (𝜑𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
7 sscrel 16397 . . . . . . 7 Rel ⊆cat
87brrelex2i 5121 . . . . . 6 (𝐻cat 𝐽𝐽 ∈ V)
9 dmexg 7047 . . . . . 6 (𝐽 ∈ V → dom 𝐽 ∈ V)
10 dmexg 7047 . . . . . 6 (dom 𝐽 ∈ V → dom dom 𝐽 ∈ V)
113, 8, 9, 104syl 19 . . . . 5 (𝜑 → dom dom 𝐽 ∈ V)
124, 6, 11isssc 16404 . . . 4 (𝜑 → (𝐻cat 𝐽 ↔ (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
133, 12mpbid 222 . . 3 (𝜑 → (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
1413simprd 479 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
15 oveq1 6614 . . . 4 (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16 oveq1 6614 . . . 4 (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦))
1715, 16sseq12d 3615 . . 3 (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦)))
18 oveq2 6615 . . . 4 (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
19 oveq2 6615 . . . 4 (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌))
2018, 19sseq12d 3615 . . 3 (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)))
2117, 20rspc2va 3308 . 2 (((𝑋𝑆𝑌𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
221, 2, 14, 21syl21anc 1322 1 (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3556   class class class wbr 4615   × cxp 5074  dom cdm 5076   Fn wfn 5844  (class class class)co 6607  cat cssc 16391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-ixp 7856  df-ssc 16394
This theorem is referenced by:  ssctr  16409  ssceq  16410  subcss2  16427
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