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Theorem fullfo 14101
 Description: The morphism map of a full functor is a surjection. (Contributed by Mario Carneiro, 27-Jan-2017.)
Hypotheses
Ref Expression
isfull.b
isfull.j
isfull.h
fullfo.f Full
fullfo.x
fullfo.y
Assertion
Ref Expression
fullfo

Proof of Theorem fullfo
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fullfo.f . . 3 Full
2 isfull.b . . . . 5
3 isfull.j . . . . 5
4 isfull.h . . . . 5
52, 3, 4isfull2 14100 . . . 4 Full
65simprbi 451 . . 3 Full
71, 6syl 16 . 2
8 fullfo.x . . 3
9 fullfo.y . . . . 5
109adantr 452 . . . 4
11 simplr 732 . . . . . 6
12 simpr 448 . . . . . 6
1311, 12oveq12d 6091 . . . . 5
1411, 12oveq12d 6091 . . . . 5
1511fveq2d 5724 . . . . . 6
1612fveq2d 5724 . . . . . 6
1715, 16oveq12d 6091 . . . . 5
1813, 14, 17foeq123d 5662 . . . 4
1910, 18rspcdv 3047 . . 3
208, 19rspcimdv 3045 . 2
217, 20mpd 15 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697   class class class wbr 4204  wfo 5444  cfv 5446  (class class class)co 6073  cbs 13461   chom 13532   cfunc 14043   Full cful 14091 This theorem is referenced by:  fulli  14102  ffthf1o  14108  fulloppc  14111  cofull  14123 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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  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-reu 2704  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-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  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-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012  df-ixp 7056  df-func 14047  df-full 14093
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