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Theorem setcval 14222
 Description: Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcval.c
setcval.u
setcval.h
setcval.o
Assertion
Ref Expression
setcval comp
Distinct variable groups:   ,,,,,   ,,,,   ,,,,
Allowed substitution hints:   (,)   (,,,,,)   (,,,,,)   (,)   (,,,,,)   (,,,,,)

Proof of Theorem setcval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 setcval.c . 2
2 df-setc 14221 . . . 4 comp
32a1i 11 . . 3 comp
4 simpr 448 . . . . 5
54opeq2d 3983 . . . 4
6 eqidd 2436 . . . . . . 7
74, 4, 6mpt2eq123dv 6128 . . . . . 6
8 setcval.h . . . . . . 7
98adantr 452 . . . . . 6
107, 9eqtr4d 2470 . . . . 5
1110opeq2d 3983 . . . 4
124, 4xpeq12d 4895 . . . . . . 7
13 eqidd 2436 . . . . . . 7
1412, 4, 13mpt2eq123dv 6128 . . . . . 6
15 setcval.o . . . . . . 7
1615adantr 452 . . . . . 6
1714, 16eqtr4d 2470 . . . . 5
1817opeq2d 3983 . . . 4 comp comp
195, 11, 18tpeq123d 3890 . . 3 comp comp
20 setcval.u . . . 4
21 elex 2956 . . . 4
2220, 21syl 16 . . 3
23 tpex 4700 . . . 4 comp
2423a1i 11 . . 3 comp
253, 19, 22, 24fvmptd 5802 . 2 comp
261, 25syl5eq 2479 1 comp
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  cvv 2948  ctp 3808  cop 3809   cmpt 4258   cxp 4868   ccom 4874  cfv 5446  (class class class)co 6073   cmpt2 6075  c1st 6339  c2nd 6340   cmap 7010  cnx 13456  cbs 13459   chom 13530  compcco 13531  csetc 14220 This theorem is referenced by:  setcbas  14223  setchomfval  14224  setccofval  14227 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-13 1727  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  ax-un 4693 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-csb 3244  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-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  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-oprab 6077  df-mpt2 6078  df-setc 14221
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