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Theorem caoftrn 6556
 Description: Transfer a transitivity law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
caofref.1
caofref.2
caofcom.3
caofass.4
caoftrn.5
Assertion
Ref Expression
caoftrn
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem caoftrn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 caoftrn.5 . . . . . 6
21ralrimivvva 2863 . . . . 5
32adantr 465 . . . 4
4 caofref.2 . . . . . 6
54ffvelrnda 6012 . . . . 5
6 caofcom.3 . . . . . 6
76ffvelrnda 6012 . . . . 5
8 caofass.4 . . . . . 6
98ffvelrnda 6012 . . . . 5
10 breq1 4436 . . . . . . . 8
1110anbi1d 704 . . . . . . 7
12 breq1 4436 . . . . . . 7
1311, 12imbi12d 320 . . . . . 6
14 breq2 4437 . . . . . . . 8
15 breq1 4436 . . . . . . . 8
1614, 15anbi12d 710 . . . . . . 7
1716imbi1d 317 . . . . . 6
18 breq2 4437 . . . . . . . 8
1918anbi2d 703 . . . . . . 7
20 breq2 4437 . . . . . . 7
2119, 20imbi12d 320 . . . . . 6
2213, 17, 21rspc3v 3206 . . . . 5
235, 7, 9, 22syl3anc 1227 . . . 4
243, 23mpd 15 . . 3
2524ralimdva 2849 . 2
26 ffn 5717 . . . . . 6
274, 26syl 16 . . . . 5
28 ffn 5717 . . . . . 6
296, 28syl 16 . . . . 5
30 caofref.1 . . . . 5
31 inidm 3689 . . . . 5
32 eqidd 2442 . . . . 5
33 eqidd 2442 . . . . 5
3427, 29, 30, 30, 31, 32, 33ofrfval 6529 . . . 4
35 ffn 5717 . . . . . 6
368, 35syl 16 . . . . 5
37 eqidd 2442 . . . . 5
3829, 36, 30, 30, 31, 33, 37ofrfval 6529 . . . 4
3934, 38anbi12d 710 . . 3
40 r19.26 2968 . . 3
4139, 40syl6bbr 263 . 2
4227, 36, 30, 30, 31, 32, 37ofrfval 6529 . 2
4325, 41, 423imtr4d 268 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   w3a 972   wceq 1381   wcel 1802  wral 2791   class class class wbr 4433   wfn 5569  wf 5570  cfv 5574   cofr 6520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pr 4672 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ofr 6522 This theorem is referenced by:  gsumbagdiaglem  17895  itg2le  22012
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