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Mirrors > Home > ILE Home > Th. List > funbrfv | Unicode version |
Description: The second argument of a binary relation on a function is the function's value. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
funbrfv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funrel 5110 | . . . 4 | |
2 | brrelex2 4550 | . . . 4 | |
3 | 1, 2 | sylan 281 | . . 3 |
4 | breq2 3903 | . . . . . 6 | |
5 | 4 | anbi2d 459 | . . . . 5 |
6 | eqeq2 2127 | . . . . 5 | |
7 | 5, 6 | imbi12d 233 | . . . 4 |
8 | funeu 5118 | . . . . . 6 | |
9 | tz6.12-1 5416 | . . . . . 6 | |
10 | 8, 9 | sylan2 284 | . . . . 5 |
11 | 10 | anabss7 557 | . . . 4 |
12 | 7, 11 | vtoclg 2720 | . . 3 |
13 | 3, 12 | mpcom 36 | . 2 |
14 | 13 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 weu 1977 cvv 2660 class class class wbr 3899 wrel 4514 wfun 5087 cfv 5093 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 |
This theorem is referenced by: funopfv 5429 fnbrfvb 5430 fvelima 5441 fvi 5446 fmptco 5554 fliftfun 5665 fliftval 5669 tfrlem5 6179 sum0 11125 isumz 11126 fsumsersdc 11132 isumclim 11158 dvaddxx 12763 dvmulxx 12764 dvcj 12769 dvrecap 12773 dvef 12783 pilem3 12791 |
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