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Mirrors > Home > ILE Home > Th. List > caofrss | Unicode version |
Description: Transfer a relation subset law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
caofref.1 | |
caofref.2 | |
caofcom.3 | |
caofrss.4 |
Ref | Expression |
---|---|
caofrss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | caofref.2 | . . . . 5 | |
2 | 1 | ffvelrnda 5603 | . . . 4 |
3 | caofcom.3 | . . . . 5 | |
4 | 3 | ffvelrnda 5603 | . . . 4 |
5 | caofrss.4 | . . . . . 6 | |
6 | 5 | ralrimivva 2539 | . . . . 5 |
7 | 6 | adantr 274 | . . . 4 |
8 | breq1 3969 | . . . . . 6 | |
9 | breq1 3969 | . . . . . 6 | |
10 | 8, 9 | imbi12d 233 | . . . . 5 |
11 | breq2 3970 | . . . . . 6 | |
12 | breq2 3970 | . . . . . 6 | |
13 | 11, 12 | imbi12d 233 | . . . . 5 |
14 | 10, 13 | rspc2va 2830 | . . . 4 |
15 | 2, 4, 7, 14 | syl21anc 1219 | . . 3 |
16 | 15 | ralimdva 2524 | . 2 |
17 | ffn 5320 | . . . 4 | |
18 | 1, 17 | syl 14 | . . 3 |
19 | ffn 5320 | . . . 4 | |
20 | 3, 19 | syl 14 | . . 3 |
21 | caofref.1 | . . 3 | |
22 | inidm 3316 | . . 3 | |
23 | eqidd 2158 | . . 3 | |
24 | eqidd 2158 | . . 3 | |
25 | 18, 20, 21, 21, 22, 23, 24 | ofrfval 6041 | . 2 |
26 | 18, 20, 21, 21, 22, 23, 24 | ofrfval 6041 | . 2 |
27 | 16, 25, 26 | 3imtr4d 202 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1335 wcel 2128 wral 2435 class class class wbr 3966 wfn 5166 wf 5167 cfv 5171 cofr 6032 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-coll 4080 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-iun 3852 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-f1 5176 df-fo 5177 df-f1o 5178 df-fv 5179 df-ofr 6034 |
This theorem is referenced by: (None) |
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