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Theorem ssc1 17079
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 17071 . . . . . . 7 Rel ⊆cat
54brrelex2i 5602 . . . . . 6 (𝐻cat 𝐽𝐽 ∈ V)
61, 5syl 17 . . . . 5 (𝜑𝐽 ∈ V)
73ssclem 17077 . . . . 5 (𝜑 → (𝐽 ∈ V ↔ 𝑇 ∈ V))
86, 7mpbid 233 . . . 4 (𝜑𝑇 ∈ V)
92, 3, 8isssc 17078 . . 3 (𝜑 → (𝐻cat 𝐽 ↔ (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
101, 9mpbid 233 . 2 (𝜑 → (𝑆𝑇 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
1110simpld 495 1 (𝜑𝑆𝑇)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  wral 3135  Vcvv 3492  wss 3933   class class class wbr 5057   × cxp 5546   Fn wfn 6343  (class class class)co 7145  cat cssc 17065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-ixp 8450  df-ssc 17068
This theorem is referenced by:  ssctr  17083  ssceq  17084  subcss1  17100  issubc3  17107  subsubc  17111
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