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Theorem subcss2 17792
Description: The morphisms of a subcategory are a subset of the morphisms of the original. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
subcss1.1 (𝜑𝐽 ∈ (Subcat‘𝐶))
subcss1.2 (𝜑𝐽 Fn (𝑆 × 𝑆))
subcss2.h 𝐻 = (Hom ‘𝐶)
subcss2.x (𝜑𝑋𝑆)
subcss2.y (𝜑𝑌𝑆)
Assertion
Ref Expression
subcss2 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))

Proof of Theorem subcss2
StepHypRef Expression
1 subcss1.2 . . 3 (𝜑𝐽 Fn (𝑆 × 𝑆))
2 subcss1.1 . . . 4 (𝜑𝐽 ∈ (Subcat‘𝐶))
3 eqid 2724 . . . 4 (Homf𝐶) = (Homf𝐶)
42, 3subcssc 17789 . . 3 (𝜑𝐽cat (Homf𝐶))
5 subcss2.x . . 3 (𝜑𝑋𝑆)
6 subcss2.y . . 3 (𝜑𝑌𝑆)
71, 4, 5, 6ssc2 17768 . 2 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋(Homf𝐶)𝑌))
8 eqid 2724 . . 3 (Base‘𝐶) = (Base‘𝐶)
9 subcss2.h . . 3 𝐻 = (Hom ‘𝐶)
102, 1, 8subcss1 17791 . . . 4 (𝜑𝑆 ⊆ (Base‘𝐶))
1110, 5sseldd 3975 . . 3 (𝜑𝑋 ∈ (Base‘𝐶))
1210, 6sseldd 3975 . . 3 (𝜑𝑌 ∈ (Base‘𝐶))
133, 8, 9, 11, 12homfval 17635 . 2 (𝜑 → (𝑋(Homf𝐶)𝑌) = (𝑋𝐻𝑌))
147, 13sseqtrd 4014 1 (𝜑 → (𝑋𝐽𝑌) ⊆ (𝑋𝐻𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wss 3940   × cxp 5664   Fn wfn 6528  cfv 6533  (class class class)co 7401  Basecbs 17143  Hom chom 17207  Homf chomf 17609  Subcatcsubc 17755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-pm 8819  df-ixp 8888  df-homf 17613  df-ssc 17756  df-subc 17758
This theorem is referenced by:  subccatid  17795  funcres  17845  funcres2b  17846  subthinc  47848
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