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Theorem cidpropd 13928
 Description: Two structures with the same base, hom-sets and composition operation have the same identity function. (Contributed by Mario Carneiro, 17-Jan-2017.)
Hypotheses
Ref Expression
catpropd.1 f f
catpropd.2 compf compf
catpropd.3
catpropd.4
Assertion
Ref Expression
cidpropd

Proof of Theorem cidpropd
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 catpropd.1 . . . . . 6 f f
21homfeqbas 13914 . . . . 5
32adantr 452 . . . 4
4 eqid 2435 . . . . . . . . . 10
5 eqid 2435 . . . . . . . . . 10
6 eqid 2435 . . . . . . . . . 10
71ad4antr 713 . . . . . . . . . 10 f f
8 simpr 448 . . . . . . . . . 10
9 simpllr 736 . . . . . . . . . 10
104, 5, 6, 7, 8, 9homfeqval 13915 . . . . . . . . 9
11 eqid 2435 . . . . . . . . . . 11 comp comp
12 eqid 2435 . . . . . . . . . . 11 comp comp
131ad5antr 715 . . . . . . . . . . 11 f f
14 catpropd.2 . . . . . . . . . . . 12 compf compf
1514ad5antr 715 . . . . . . . . . . 11 compf compf
16 simplr 732 . . . . . . . . . . 11
17 simp-4r 744 . . . . . . . . . . 11
18 simpr 448 . . . . . . . . . . 11
19 simpllr 736 . . . . . . . . . . 11
204, 5, 11, 12, 13, 15, 16, 17, 17, 18, 19comfeqval 13926 . . . . . . . . . 10 comp comp
2120eqeq1d 2443 . . . . . . . . 9 comp comp
2210, 21raleqbidva 2910 . . . . . . . 8 comp comp
234, 5, 6, 7, 9, 8homfeqval 13915 . . . . . . . . 9
247adantr 452 . . . . . . . . . . 11 f f
2514ad5antr 715 . . . . . . . . . . 11 compf compf
269adantr 452 . . . . . . . . . . 11
27 simplr 732 . . . . . . . . . . 11
28 simpllr 736 . . . . . . . . . . 11
29 simpr 448 . . . . . . . . . . 11
304, 5, 11, 12, 24, 25, 26, 26, 27, 28, 29comfeqval 13926 . . . . . . . . . 10 comp comp
3130eqeq1d 2443 . . . . . . . . 9 comp comp
3223, 31raleqbidva 2910 . . . . . . . 8 comp comp
3322, 32anbi12d 692 . . . . . . 7 comp comp comp comp
3433ralbidva 2713 . . . . . 6 comp comp comp comp
3534riotabidva 6558 . . . . 5 comp comp comp comp
361ad2antrr 707 . . . . . . 7 f f
37 simpr 448 . . . . . . 7
384, 5, 6, 36, 37, 37homfeqval 13915 . . . . . 6
392ad2antrr 707 . . . . . . 7
4039raleqdv 2902 . . . . . 6 comp comp comp comp
4138, 40riotaeqbidv 6544 . . . . 5 comp comp comp comp
4235, 41eqtrd 2467 . . . 4 comp comp comp comp
433, 42mpteq12dva 4278 . . 3 comp comp comp comp
44 simpr 448 . . . 4
45 eqid 2435 . . . 4
464, 5, 11, 44, 45cidfval 13893 . . 3 comp comp
47 eqid 2435 . . . 4
48 catpropd.3 . . . . . 6
49 catpropd.4 . . . . . 6
501, 14, 48, 49catpropd 13927 . . . . 5
5150biimpa 471 . . . 4
52 eqid 2435 . . . 4
5347, 6, 12, 51, 52cidfval 13893 . . 3 comp comp
5443, 46, 533eqtr4d 2477 . 2
55 simpr 448 . . . . 5
56 cidffn 13895 . . . . . . 7
57 fndm 5536 . . . . . . 7
5856, 57ax-mp 8 . . . . . 6
5958eleq2i 2499 . . . . 5
6055, 59sylnibr 297 . . . 4
61 ndmfv 5747 . . . 4
6260, 61syl 16 . . 3
6358eleq2i 2499 . . . . . . 7
6450, 63syl6bbr 255 . . . . . 6
6564notbid 286 . . . . 5
6665biimpa 471 . . . 4
67 ndmfv 5747 . . . 4
6866, 67syl 16 . . 3
6962, 68eqtr4d 2470 . 2
7054, 69pm2.61dan 767 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697  c0 3620  cop 3809   cmpt 4258   cdm 4870   wfn 5441  cfv 5446  (class class class)co 6073  crio 6534  cbs 13461   chom 13532  compcco 13533  ccat 13881  ccid 13882   f chomf 13883  compfccomf 13884 This theorem is referenced by:  funcpropd  14089  curfpropd  14322 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-riota 6541  df-cat 13885  df-cid 13886  df-homf 13887  df-comf 13888
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