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Theorem ssc2 16703
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 2761 . . . . . 6 (𝜑 → dom dom 𝐽 = dom dom 𝐽)
63, 5sscfn2 16699 . . . . 5 (𝜑𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
7 sscrel 16694 . . . . . . 7 Rel ⊆cat
87brrelex2i 5316 . . . . . 6 (𝐻cat 𝐽𝐽 ∈ V)
9 dmexg 7263 . . . . . 6 (𝐽 ∈ V → dom 𝐽 ∈ V)
10 dmexg 7263 . . . . . 6 (dom 𝐽 ∈ V → dom dom 𝐽 ∈ V)
113, 8, 9, 104syl 19 . . . . 5 (𝜑 → dom dom 𝐽 ∈ V)
124, 6, 11isssc 16701 . . . 4 (𝜑 → (𝐻cat 𝐽 ↔ (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
133, 12mpbid 222 . . 3 (𝜑 → (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
1413simprd 482 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
15 oveq1 6821 . . . 4 (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16 oveq1 6821 . . . 4 (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦))
1715, 16sseq12d 3775 . . 3 (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦)))
18 oveq2 6822 . . . 4 (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
19 oveq2 6822 . . . 4 (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌))
2018, 19sseq12d 3775 . . 3 (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)))
2117, 20rspc2va 3462 . 2 (((𝑋𝑆𝑌𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
221, 2, 14, 21syl21anc 1476 1 (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  wss 3715   class class class wbr 4804   × cxp 5264  dom cdm 5266   Fn wfn 6044  (class class class)co 6814  cat cssc 16688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-ixp 8077  df-ssc 16691
This theorem is referenced by:  ssctr  16706  ssceq  16707  subcss2  16724
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