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Theorem ssc1 17865
Description: Infer subset relation on objects from the subcategory subset relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
isssc.1 (𝜑𝐻 Fn (𝑆 × 𝑆))
isssc.2 (𝜑𝐽 Fn (𝑇 × 𝑇))
ssc1.3 (𝜑𝐻cat 𝐽)
Assertion
Ref Expression
ssc1 (𝜑𝑆𝑇)

Proof of Theorem ssc1
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssc1.3 . . 3 (𝜑𝐻cat 𝐽)
2 isssc.1 . . . 4 (𝜑𝐻 Fn (𝑆 × 𝑆))
3 isssc.2 . . . 4 (𝜑𝐽 Fn (𝑇 × 𝑇))
4 sscrel 17857 . . . . . . 7 Rel ⊆cat
54brrelex2i 5742 . . . . . 6 (𝐻cat 𝐽𝐽 ∈ V)
61, 5syl 17 . . . . 5 (𝜑𝐽 ∈ V)
73ssclem 17863 . . . . 5 (𝜑 → (𝐽 ∈ V ↔ 𝑇 ∈ V))
86, 7mpbid 232 . . . 4 (𝜑𝑇 ∈ V)
92, 3, 8isssc 17864 . . 3 (𝜑 → (𝐻cat 𝐽 ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
101, 9mpbid 232 . 2 (𝜑 → (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
1110simpld 494 1 (𝜑𝑆𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  Vcvv 3480  wss 3951   class class class wbr 5143   × cxp 5683   Fn wfn 6556  (class class class)co 7431  cat cssc 17851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-ixp 8938  df-ssc 17854
This theorem is referenced by:  ssctr  17869  ssceq  17870  subcss1  17887  issubc3  17894  subsubc  17898
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