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

Dummy variable is distinct from all other variables.
StepHypRef Expression
1 caofref.2 . . . . 5
21ffvelrnda 6032 . . . 4
3 caofcom.3 . . . . 5
43ffvelrnda 6032 . . . 4
5 caofrss.4 . . . . . 6
65ralrimivva 2878 . . . . 5
76adantr 465 . . . 4
8 breq1 4459 . . . . . 6
9 breq1 4459 . . . . . 6
108, 9imbi12d 320 . . . . 5
11 breq2 4460 . . . . . 6
12 breq2 4460 . . . . . 6
1311, 12imbi12d 320 . . . . 5
1410, 13rspc2va 3220 . . . 4
152, 4, 7, 14syl21anc 1227 . . 3
1615ralimdva 2865 . 2
17 ffn 5737 . . . 4
181, 17syl 16 . . 3
19 ffn 5737 . . . 4
203, 19syl 16 . . 3
21 caofref.1 . . 3
22 inidm 3703 . . 3
23 eqidd 2458 . . 3
24 eqidd 2458 . . 3
2518, 20, 21, 21, 22, 23, 24ofrfval 6547 . 2
2618, 20, 21, 21, 22, 23, 24ofrfval 6547 . 2
2716, 25, 263imtr4d 268 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395   wcel 1819  wral 2807   class class class wbr 4456   wfn 5589  wf 5590  cfv 5594   cofr 6538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ofr 6540 This theorem is referenced by: (None)
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