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Theorem cocan2 5371
Description: A surjection is right-cancelable. (Contributed by FL, 21-Nov-2011.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
cocan2  F : -onto->  H  Fn  K  Fn  H  o.  F  K  o.  F  H  K

Proof of Theorem cocan2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fof 5049 . . . . . . 7  F : -onto->  F : -->
213ad2ant1 924 . . . . . 6  F : -onto->  H  Fn  K  Fn  F : -->
3 fvco3 5187 . . . . . 6  F : -->  H  o.  F `  H `  F `
42, 3sylan 267 . . . . 5  F : -onto->  H  Fn  K  Fn  H  o.  F `  H `  F `
5 fvco3 5187 . . . . . 6  F : -->  K  o.  F `  K `  F `
62, 5sylan 267 . . . . 5  F : -onto->  H  Fn  K  Fn  K  o.  F `  K `  F `
74, 6eqeq12d 2051 . . . 4  F : -onto->  H  Fn  K  Fn  H  o.  F `
 K  o.  F `  H `  F `  K `  F `
87ralbidva 2316 . . 3  F : -onto->  H  Fn  K  Fn  H  o.  F `  K  o.  F `  H `  F `  K `
 F `
9 fveq2 5121 . . . . . 6  F `  H `  F `  H `
10 fveq2 5121 . . . . . 6  F `  K `  F `  K `
119, 10eqeq12d 2051 . . . . 5  F `  H `  F `
 K `  F `  H `  K `
1211cbvfo 5368 . . . 4  F : -onto->  H `  F `
 K `  F `  H `  K `
13123ad2ant1 924 . . 3  F : -onto->  H  Fn  K  Fn  H `  F `  K `
 F `  H `  K `
148, 13bitrd 177 . 2  F : -onto->  H  Fn  K  Fn  H  o.  F `  K  o.  F `  H `  K `
15 simp2 904 . . . 4  F : -onto->  H  Fn  K  Fn  H  Fn
16 fnfco 5008 . . . 4  H  Fn  F : -->  H  o.  F  Fn
1715, 2, 16syl2anc 391 . . 3  F : -onto->  H  Fn  K  Fn  H  o.  F  Fn
18 simp3 905 . . . 4  F : -onto->  H  Fn  K  Fn  K  Fn
19 fnfco 5008 . . . 4  K  Fn  F : -->  K  o.  F  Fn
2018, 2, 19syl2anc 391 . . 3  F : -onto->  H  Fn  K  Fn  K  o.  F  Fn
21 eqfnfv 5208 . . 3  H  o.  F  Fn  K  o.  F  Fn  H  o.  F  K  o.  F  H  o.  F `  K  o.  F `
2217, 20, 21syl2anc 391 . 2  F : -onto->  H  Fn  K  Fn  H  o.  F  K  o.  F  H  o.  F `  K  o.  F `
23 eqfnfv 5208 . . 3  H  Fn  K  Fn  H  K  H `  K `
2415, 18, 23syl2anc 391 . 2  F : -onto->  H  Fn  K  Fn  H  K  H `  K `
2514, 22, 243bitr4d 209 1  F : -onto->  H  Fn  K  Fn  H  o.  F  K  o.  F  H  K
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wceq 1242   wcel 1390  wral 2300    o. ccom 4292    Fn wfn 4840   -->wf 4841   -onto->wfo 4843   ` cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853
This theorem is referenced by: (None)
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