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Theorem rescval 14019
 Description: Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypothesis
Ref Expression
rescval.1 cat
Assertion
Ref Expression
rescval s sSet

Proof of Theorem rescval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rescval.1 . 2 cat
2 elex 2956 . . 3
3 elex 2956 . . 3
4 simpl 444 . . . . . 6
5 simpr 448 . . . . . . . 8
65dmeqd 5064 . . . . . . 7
76dmeqd 5064 . . . . . 6
84, 7oveq12d 6091 . . . . 5 s s
95opeq2d 3983 . . . . 5
108, 9oveq12d 6091 . . . 4 s sSet s sSet
11 df-resc 14003 . . . 4 cat s sSet
12 ovex 6098 . . . 4 s sSet
1310, 11, 12ovmpt2a 6196 . . 3 cat s sSet
142, 3, 13syl2an 464 . 2 cat s sSet
151, 14syl5eq 2479 1 s sSet
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cvv 2948  cop 3809   cdm 4870  cfv 5446  (class class class)co 6073  cnx 13458   sSet csts 13459   ↾s cress 13462   chom 13532   cat cresc 14000 This theorem is referenced by:  rescval2  14020 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-resc 14003
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