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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 5521 . . . 4
3 caofcom.3 . . . . 5
43ffvelrnda 5521 . . . 4
5 caofrss.4 . . . . . 6
65ralrimivva 2489 . . . . 5
76adantr 272 . . . 4
8 breq1 3900 . . . . . 6
9 breq1 3900 . . . . . 6
108, 9imbi12d 233 . . . . 5
11 breq2 3901 . . . . . 6
12 breq2 3901 . . . . . 6
1311, 12imbi12d 233 . . . . 5
1410, 13rspc2va 2775 . . . 4
152, 4, 7, 14syl21anc 1198 . . 3
1615ralimdva 2474 . 2
17 ffn 5240 . . . 4
181, 17syl 14 . . 3
19 ffn 5240 . . . 4
203, 19syl 14 . . 3
21 caofref.1 . . 3
22 inidm 3253 . . 3
23 eqidd 2116 . . 3
24 eqidd 2116 . . 3
2518, 20, 21, 21, 22, 23, 24ofrfval 5956 . 2
2618, 20, 21, 21, 22, 23, 24ofrfval 5956 . 2
2716, 25, 263imtr4d 202 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1314   wcel 1463  wral 2391   class class class wbr 3897   wfn 5086  wf 5087  cfv 5091   cofr 5947 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ofr 5949 This theorem is referenced by: (None)
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