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Theorem ssc2 16305
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 2611 . . . . . 6 (𝜑 → dom dom 𝐽 = dom dom 𝐽)
63, 5sscfn2 16301 . . . . 5 (𝜑𝐽 Fn (dom dom 𝐽 × dom dom 𝐽))
7 sscrel 16296 . . . . . . 7 Rel ⊆cat
87brrelex2i 5083 . . . . . 6 (𝐻cat 𝐽𝐽 ∈ V)
9 dmexg 6989 . . . . . 6 (𝐽 ∈ V → dom 𝐽 ∈ V)
10 dmexg 6989 . . . . . 6 (dom 𝐽 ∈ V → dom dom 𝐽 ∈ V)
113, 8, 9, 104syl 19 . . . . 5 (𝜑 → dom dom 𝐽 ∈ V)
124, 6, 11isssc 16303 . . . 4 (𝜑 → (𝐻cat 𝐽 ↔ (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))))
133, 12mpbid 221 . . 3 (𝜑 → (𝑆 ⊆ dom dom 𝐽 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)))
1413simprd 478 . 2 (𝜑 → ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦))
15 oveq1 6556 . . . 4 (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
16 oveq1 6556 . . . 4 (𝑥 = 𝑋 → (𝑥𝐽𝑦) = (𝑋𝐽𝑦))
1715, 16sseq12d 3597 . . 3 (𝑥 = 𝑋 → ((𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦) ↔ (𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦)))
18 oveq2 6557 . . . 4 (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
19 oveq2 6557 . . . 4 (𝑦 = 𝑌 → (𝑋𝐽𝑦) = (𝑋𝐽𝑌))
2018, 19sseq12d 3597 . . 3 (𝑦 = 𝑌 → ((𝑋𝐻𝑦) ⊆ (𝑋𝐽𝑦) ↔ (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌)))
2117, 20rspc2va 3294 . 2 (((𝑋𝑆𝑌𝑆) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥𝐻𝑦) ⊆ (𝑥𝐽𝑦)) → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
221, 2, 14, 21syl21anc 1317 1 (𝜑 → (𝑋𝐻𝑌) ⊆ (𝑋𝐽𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  wss 3540   class class class wbr 4583   × cxp 5036  dom cdm 5038   Fn wfn 5799  (class class class)co 6549  cat cssc 16290
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-ixp 7795  df-ssc 16293
This theorem is referenced by:  ssctr  16308  ssceq  16309  subcss2  16326
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