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Theorem ssc2 16878
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 2779 . . . . . 6 (𝜑 → dom dom 𝐽 = dom dom 𝐽)
63, 5sscfn2 16874 . . . . 5 (𝜑𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
7 sscrel 16869 . . . . . . 7 Rel ⊆cat
87brrelex2i 5409 . . . . . 6 (𝐻cat 𝐽𝐽 ∈ V)
9 dmexg 7377 . . . . . 6 (𝐽 ∈ V → dom 𝐽 ∈ V)
10 dmexg 7377 . . . . . 6 (dom 𝐽 ∈ V → dom dom 𝐽 ∈ V)
113, 8, 9, 104syl 19 . . . . 5 (𝜑 → dom dom 𝐽 ∈ V)
124, 6, 11isssc 16876 . . . 4 (𝜑 → (𝐻cat 𝐽 ↔ (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
133, 12mpbid 224 . . 3 (𝜑 → (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
1413simprd 491 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
15 oveq1 6931 . . . 4 (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16 oveq1 6931 . . . 4 (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦))
1715, 16sseq12d 3853 . . 3 (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦)))
18 oveq2 6932 . . . 4 (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
19 oveq2 6932 . . . 4 (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌))
2018, 19sseq12d 3853 . . 3 (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)))
2117, 20rspc2va 3525 . 2 (((𝑋𝑆𝑌𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
221, 2, 14, 21syl21anc 828 1 (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wral 3090  Vcvv 3398  wss 3792   class class class wbr 4888   × cxp 5355  dom cdm 5357   Fn wfn 6132  (class class class)co 6924  cat cssc 16863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-ixp 8197  df-ssc 16866
This theorem is referenced by:  ssctr  16881  ssceq  16882  subcss2  16899
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