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Theorem subcss2 17799
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 (πœ‘ β†’ 𝐽 ∈ (Subcatβ€˜πΆ))
subcss1.2 (πœ‘ β†’ 𝐽 Fn (𝑆 Γ— 𝑆))
subcss2.h 𝐻 = (Hom β€˜πΆ)
subcss2.x (πœ‘ β†’ 𝑋 ∈ 𝑆)
subcss2.y (πœ‘ β†’ π‘Œ ∈ 𝑆)
Assertion
Ref Expression
subcss2 (πœ‘ β†’ (π‘‹π½π‘Œ) βŠ† (π‘‹π»π‘Œ))

Proof of Theorem subcss2
StepHypRef Expression
1 subcss1.2 . . 3 (πœ‘ β†’ 𝐽 Fn (𝑆 Γ— 𝑆))
2 subcss1.1 . . . 4 (πœ‘ β†’ 𝐽 ∈ (Subcatβ€˜πΆ))
3 eqid 2726 . . . 4 (Homf β€˜πΆ) = (Homf β€˜πΆ)
42, 3subcssc 17796 . . 3 (πœ‘ β†’ 𝐽 βŠ†cat (Homf β€˜πΆ))
5 subcss2.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝑆)
6 subcss2.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝑆)
71, 4, 5, 6ssc2 17775 . 2 (πœ‘ β†’ (π‘‹π½π‘Œ) βŠ† (𝑋(Homf β€˜πΆ)π‘Œ))
8 eqid 2726 . . 3 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
9 subcss2.h . . 3 𝐻 = (Hom β€˜πΆ)
102, 1, 8subcss1 17798 . . . 4 (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜πΆ))
1110, 5sseldd 3978 . . 3 (πœ‘ β†’ 𝑋 ∈ (Baseβ€˜πΆ))
1210, 6sseldd 3978 . . 3 (πœ‘ β†’ π‘Œ ∈ (Baseβ€˜πΆ))
133, 8, 9, 11, 12homfval 17642 . 2 (πœ‘ β†’ (𝑋(Homf β€˜πΆ)π‘Œ) = (π‘‹π»π‘Œ))
147, 13sseqtrd 4017 1 (πœ‘ β†’ (π‘‹π½π‘Œ) βŠ† (π‘‹π»π‘Œ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098   βŠ† wss 3943   Γ— cxp 5667   Fn wfn 6531  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150  Hom chom 17214  Homf chomf 17616  Subcatcsubc 17762
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 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-pm 8822  df-ixp 8891  df-homf 17620  df-ssc 17763  df-subc 17765
This theorem is referenced by:  subccatid  17802  funcres  17852  funcres2b  17853  subthinc  47916
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