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Mirrors > Home > ILE Home > Th. List > caofref | Unicode version |
Description: Transfer a reflexive law to the function relation. (Contributed by Mario Carneiro, 28-Jul-2014.) |
Ref | Expression |
---|---|
caofref.1 | |
caofref.2 | |
caofref.3 |
Ref | Expression |
---|---|
caofref |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . . . 5 | |
2 | 1, 1 | breq12d 3937 | . . . 4 |
3 | caofref.3 | . . . . . 6 | |
4 | 3 | ralrimiva 2503 | . . . . 5 |
5 | 4 | adantr 274 | . . . 4 |
6 | caofref.2 | . . . . 5 | |
7 | 6 | ffvelrnda 5548 | . . . 4 |
8 | 2, 5, 7 | rspcdva 2789 | . . 3 |
9 | 8 | ralrimiva 2503 | . 2 |
10 | ffn 5267 | . . . 4 | |
11 | 6, 10 | syl 14 | . . 3 |
12 | caofref.1 | . . 3 | |
13 | inidm 3280 | . . 3 | |
14 | eqidd 2138 | . . 3 | |
15 | 11, 11, 12, 12, 13, 14, 14 | ofrfval 5983 | . 2 |
16 | 9, 15 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2414 class class class wbr 3924 wfn 5113 wf 5114 cfv 5118 cofr 5974 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-ofr 5976 |
This theorem is referenced by: (None) |
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