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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 3989 | . . . 4 |
3 | caofref.3 | . . . . . 6 | |
4 | 3 | ralrimiva 2537 | . . . . 5 |
5 | 4 | adantr 274 | . . . 4 |
6 | caofref.2 | . . . . 5 | |
7 | 6 | ffvelrnda 5614 | . . . 4 |
8 | 2, 5, 7 | rspcdva 2830 | . . 3 |
9 | 8 | ralrimiva 2537 | . 2 |
10 | ffn 5331 | . . . 4 | |
11 | 6, 10 | syl 14 | . . 3 |
12 | caofref.1 | . . 3 | |
13 | inidm 3326 | . . 3 | |
14 | eqidd 2165 | . . 3 | |
15 | 11, 11, 12, 12, 13, 14, 14 | ofrfval 6052 | . 2 |
16 | 9, 15 | mpbird 166 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wcel 2135 wral 2442 class class class wbr 3976 wfn 5177 wf 5178 cfv 5182 cofr 6043 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ofr 6045 |
This theorem is referenced by: (None) |
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