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Mirrors > Home > ILE Home > Th. List > fnasrn | Unicode version |
Description: A function expressed as the range of another function. (Contributed by Mario Carneiro, 22-Jun-2013.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dfmpt.1 |
Ref | Expression |
---|---|
fnasrn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfmpt.1 | . . 3 | |
2 | 1 | dfmpt 5590 | . 2 |
3 | eqid 2137 | . . . . 5 | |
4 | 3 | rnmpt 4782 | . . . 4 |
5 | velsn 3539 | . . . . . 6 | |
6 | 5 | rexbii 2440 | . . . . 5 |
7 | 6 | abbii 2253 | . . . 4 |
8 | 4, 7 | eqtr4i 2161 | . . 3 |
9 | df-iun 3810 | . . 3 | |
10 | 8, 9 | eqtr4i 2161 | . 2 |
11 | 2, 10 | eqtr4i 2161 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1331 wcel 1480 cab 2123 wrex 2415 cvv 2681 csn 3522 cop 3525 ciun 3808 cmpt 3984 crn 4535 |
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-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-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-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-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 |
This theorem is referenced by: idref 5651 |
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