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Theorem resfval 14094
 Description: Value of the functor restriction operator. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
resfval.c
resfval.d
Assertion
Ref Expression
resfval f
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem resfval
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-resf 14063 . . 3 f
21a1i 11 . 2 f
3 simprl 734 . . . . 5
43fveq2d 5735 . . . 4
5 simprr 735 . . . . . 6
65dmeqd 5075 . . . . 5
76dmeqd 5075 . . . 4
84, 7reseq12d 5150 . . 3
93fveq2d 5735 . . . . . 6
109fveq1d 5733 . . . . 5
115fveq1d 5733 . . . . 5
1210, 11reseq12d 5150 . . . 4
136, 12mpteq12dv 4290 . . 3
148, 13opeq12d 3994 . 2
15 resfval.c . . 3
16 elex 2966 . . 3
1715, 16syl 16 . 2
18 resfval.d . . 3
19 elex 2966 . . 3
2018, 19syl 16 . 2
21 opex 4430 . . 3
2221a1i 11 . 2
232, 14, 17, 20, 22ovmpt2d 6204 1 f
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  cvv 2958  cop 3819   cmpt 4269   cdm 4881   cres 4883  cfv 5457  (class class class)co 6084   cmpt2 6086  c1st 6350  c2nd 6351   f cresf 14059 This theorem is referenced by:  resfval2  14095  resf1st  14096  resf2nd  14097  funcres  14098 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-resf 14063
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