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Theorem ssc1 17869
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 17861 . . . . . . 7 Rel ⊆cat
54brrelex2i 5746 . . . . . 6 (𝐻cat 𝐽𝐽 ∈ V)
61, 5syl 17 . . . . 5 (𝜑𝐽 ∈ V)
73ssclem 17867 . . . . 5 (𝜑 → (𝐽 ∈ V ↔ 𝑇 ∈ V))
86, 7mpbid 232 . . . 4 (𝜑𝑇 ∈ V)
92, 3, 8isssc 17868 . . 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 2106  wral 3059  Vcvv 3478  wss 3963   class class class wbr 5148   × cxp 5687   Fn wfn 6558  (class class class)co 7431  cat cssc 17855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-ixp 8937  df-ssc 17858
This theorem is referenced by:  ssctr  17873  ssceq  17874  subcss1  17893  issubc3  17900  subsubc  17904
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