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Theorem caoftrn 6339
 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 2799 . . . . 5
32adantr 452 . . . 4
4 caofref.2 . . . . . 6
54ffvelrnda 5870 . . . . 5
6 caofcom.3 . . . . . 6
76ffvelrnda 5870 . . . . 5
8 caofass.4 . . . . . 6
98ffvelrnda 5870 . . . . 5
10 breq1 4215 . . . . . . . 8
1110anbi1d 686 . . . . . . 7
12 breq1 4215 . . . . . . 7
1311, 12imbi12d 312 . . . . . 6
14 breq2 4216 . . . . . . . 8
15 breq1 4215 . . . . . . . 8
1614, 15anbi12d 692 . . . . . . 7
1716imbi1d 309 . . . . . 6
18 breq2 4216 . . . . . . . 8
1918anbi2d 685 . . . . . . 7
20 breq2 4216 . . . . . . 7
2119, 20imbi12d 312 . . . . . 6
2213, 17, 21rspc3v 3061 . . . . 5
235, 7, 9, 22syl3anc 1184 . . . 4
243, 23mpd 15 . . 3
2524ralimdva 2784 . 2
26 ffn 5591 . . . . . 6
274, 26syl 16 . . . . 5
28 ffn 5591 . . . . . 6
296, 28syl 16 . . . . 5
30 caofref.1 . . . . 5
31 inidm 3550 . . . . 5
32 eqidd 2437 . . . . 5
33 eqidd 2437 . . . . 5
3427, 29, 30, 30, 31, 32, 33ofrfval 6313 . . . 4
35 ffn 5591 . . . . . 6
368, 35syl 16 . . . . 5
37 eqidd 2437 . . . . 5
3829, 36, 30, 30, 31, 33, 37ofrfval 6313 . . . 4
3934, 38anbi12d 692 . . 3
40 r19.26 2838 . . 3
4139, 40syl6bbr 255 . 2
4227, 36, 30, 30, 31, 32, 37ofrfval 6313 . 2
4325, 41, 423imtr4d 260 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   w3a 936   wceq 1652   wcel 1725  wral 2705   class class class wbr 4212   wfn 5449  wf 5450  cfv 5454   cofr 6304 This theorem is referenced by:  gsumbagdiaglem  16440  itg2le  19631 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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403 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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ofr 6306
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