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Theorem funcres2 17789
Description: A functor into a restricted category is also a functor into the whole category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
funcres2 (𝑅 ∈ (Subcatβ€˜π·) β†’ (𝐢 Func (𝐷 β†Ύcat 𝑅)) βŠ† (𝐢 Func 𝐷))

Proof of Theorem funcres2
Dummy variables 𝑓 𝑔 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relfunc 17753 . . 3 Rel (𝐢 Func (𝐷 β†Ύcat 𝑅))
21a1i 11 . 2 (𝑅 ∈ (Subcatβ€˜π·) β†’ Rel (𝐢 Func (𝐷 β†Ύcat 𝑅)))
3 simpr 486 . . . . 5 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔)
4 eqid 2733 . . . . . 6 (Baseβ€˜πΆ) = (Baseβ€˜πΆ)
5 eqid 2733 . . . . . 6 (Hom β€˜πΆ) = (Hom β€˜πΆ)
6 simpl 484 . . . . . 6 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ 𝑅 ∈ (Subcatβ€˜π·))
7 eqidd 2734 . . . . . . 7 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ dom dom 𝑅 = dom dom 𝑅)
86, 7subcfn 17732 . . . . . 6 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ 𝑅 Fn (dom dom 𝑅 Γ— dom dom 𝑅))
9 eqid 2733 . . . . . . . 8 (Baseβ€˜(𝐷 β†Ύcat 𝑅)) = (Baseβ€˜(𝐷 β†Ύcat 𝑅))
104, 9, 3funcf1 17757 . . . . . . 7 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ 𝑓:(Baseβ€˜πΆ)⟢(Baseβ€˜(𝐷 β†Ύcat 𝑅)))
11 eqid 2733 . . . . . . . . 9 (𝐷 β†Ύcat 𝑅) = (𝐷 β†Ύcat 𝑅)
12 eqid 2733 . . . . . . . . 9 (Baseβ€˜π·) = (Baseβ€˜π·)
13 subcrcl 17704 . . . . . . . . . 10 (𝑅 ∈ (Subcatβ€˜π·) β†’ 𝐷 ∈ Cat)
1413adantr 482 . . . . . . . . 9 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ 𝐷 ∈ Cat)
156, 8, 12subcss1 17733 . . . . . . . . 9 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ dom dom 𝑅 βŠ† (Baseβ€˜π·))
1611, 12, 14, 8, 15rescbas 17717 . . . . . . . 8 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ dom dom 𝑅 = (Baseβ€˜(𝐷 β†Ύcat 𝑅)))
1716feq3d 6656 . . . . . . 7 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ (𝑓:(Baseβ€˜πΆ)⟢dom dom 𝑅 ↔ 𝑓:(Baseβ€˜πΆ)⟢(Baseβ€˜(𝐷 β†Ύcat 𝑅))))
1810, 17mpbird 257 . . . . . 6 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ 𝑓:(Baseβ€˜πΆ)⟢dom dom 𝑅)
19 eqid 2733 . . . . . . . 8 (Hom β€˜(𝐷 β†Ύcat 𝑅)) = (Hom β€˜(𝐷 β†Ύcat 𝑅))
20 simplr 768 . . . . . . . 8 (((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔)
21 simprl 770 . . . . . . . 8 (((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ π‘₯ ∈ (Baseβ€˜πΆ))
22 simprr 772 . . . . . . . 8 (((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ 𝑦 ∈ (Baseβ€˜πΆ))
234, 5, 19, 20, 21, 22funcf2 17759 . . . . . . 7 (((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ (π‘₯𝑔𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((π‘“β€˜π‘₯)(Hom β€˜(𝐷 β†Ύcat 𝑅))(π‘“β€˜π‘¦)))
2411, 12, 14, 8, 15reschom 17719 . . . . . . . . . 10 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ 𝑅 = (Hom β€˜(𝐷 β†Ύcat 𝑅)))
2524adantr 482 . . . . . . . . 9 (((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ 𝑅 = (Hom β€˜(𝐷 β†Ύcat 𝑅)))
2625oveqd 7375 . . . . . . . 8 (((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ ((π‘“β€˜π‘₯)𝑅(π‘“β€˜π‘¦)) = ((π‘“β€˜π‘₯)(Hom β€˜(𝐷 β†Ύcat 𝑅))(π‘“β€˜π‘¦)))
2726feq3d 6656 . . . . . . 7 (((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ ((π‘₯𝑔𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((π‘“β€˜π‘₯)𝑅(π‘“β€˜π‘¦)) ↔ (π‘₯𝑔𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((π‘“β€˜π‘₯)(Hom β€˜(𝐷 β†Ύcat 𝑅))(π‘“β€˜π‘¦))))
2823, 27mpbird 257 . . . . . 6 (((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) ∧ (π‘₯ ∈ (Baseβ€˜πΆ) ∧ 𝑦 ∈ (Baseβ€˜πΆ))) β†’ (π‘₯𝑔𝑦):(π‘₯(Hom β€˜πΆ)𝑦)⟢((π‘“β€˜π‘₯)𝑅(π‘“β€˜π‘¦)))
294, 5, 6, 8, 18, 28funcres2b 17788 . . . . 5 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ (𝑓(𝐢 Func 𝐷)𝑔 ↔ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔))
303, 29mpbird 257 . . . 4 ((𝑅 ∈ (Subcatβ€˜π·) ∧ 𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔) β†’ 𝑓(𝐢 Func 𝐷)𝑔)
3130ex 414 . . 3 (𝑅 ∈ (Subcatβ€˜π·) β†’ (𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔 β†’ 𝑓(𝐢 Func 𝐷)𝑔))
32 df-br 5107 . . 3 (𝑓(𝐢 Func (𝐷 β†Ύcat 𝑅))𝑔 ↔ βŸ¨π‘“, π‘”βŸ© ∈ (𝐢 Func (𝐷 β†Ύcat 𝑅)))
33 df-br 5107 . . 3 (𝑓(𝐢 Func 𝐷)𝑔 ↔ βŸ¨π‘“, π‘”βŸ© ∈ (𝐢 Func 𝐷))
3431, 32, 333imtr3g 295 . 2 (𝑅 ∈ (Subcatβ€˜π·) β†’ (βŸ¨π‘“, π‘”βŸ© ∈ (𝐢 Func (𝐷 β†Ύcat 𝑅)) β†’ βŸ¨π‘“, π‘”βŸ© ∈ (𝐢 Func 𝐷)))
352, 34relssdv 5745 1 (𝑅 ∈ (Subcatβ€˜π·) β†’ (𝐢 Func (𝐷 β†Ύcat 𝑅)) βŠ† (𝐢 Func 𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   βŠ† wss 3911  βŸ¨cop 4593   class class class wbr 5106  dom cdm 5634  Rel wrel 5639  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  Hom chom 17149  Catccat 17549   β†Ύcat cresc 17696  Subcatcsubc 17697   Func cfunc 17745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-map 8770  df-pm 8771  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-hom 17162  df-cco 17163  df-cat 17553  df-cid 17554  df-homf 17555  df-ssc 17698  df-resc 17699  df-subc 17700  df-func 17749
This theorem is referenced by:  fthres2  17824  ressffth  17830  funcsetcres2  17984
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