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Theorem ssc1 17091
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 17083 . . . . . . 7 Rel ⊆cat
54brrelex2i 5596 . . . . . 6 (𝐻cat 𝐽𝐽 ∈ V)
61, 5syl 17 . . . . 5 (𝜑𝐽 ∈ V)
73ssclem 17089 . . . . 5 (𝜑 → (𝐽 ∈ V ↔ 𝑇 ∈ V))
86, 7mpbid 235 . . . 4 (𝜑𝑇 ∈ V)
92, 3, 8isssc 17090 . . 3 (𝜑 → (𝐻cat 𝐽 ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
101, 9mpbid 235 . 2 (𝜑 → (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
1110simpld 498 1 (𝜑𝑆𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  wral 3133  Vcvv 3480  wss 3919   class class class wbr 5052   × cxp 5540   Fn wfn 6338  (class class class)co 7149  cat cssc 17077
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-ov 7152  df-ixp 8458  df-ssc 17080
This theorem is referenced by:  ssctr  17095  ssceq  17096  subcss1  17112  issubc3  17119  subsubc  17123
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