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Mirrors > Home > ILE Home > Th. List > fvelrnb | Unicode version |
Description: A member of a function's range is a value of the function. (Contributed by NM, 31-Oct-1995.) |
Ref | Expression |
---|---|
fvelrnb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2312 | . . . 4 | |
2 | 19.41v 1782 | . . . . 5 | |
3 | simpl 102 | . . . . . . . . . 10 | |
4 | 3 | anim1i 323 | . . . . . . . . 9 |
5 | 4 | ancomd 254 | . . . . . . . 8 |
6 | funfvex 5192 | . . . . . . . . 9 | |
7 | 6 | funfni 4999 | . . . . . . . 8 |
8 | 5, 7 | syl 14 | . . . . . . 7 |
9 | simpr 103 | . . . . . . . . 9 | |
10 | 9 | eleq1d 2106 | . . . . . . . 8 |
11 | 10 | adantr 261 | . . . . . . 7 |
12 | 8, 11 | mpbid 135 | . . . . . 6 |
13 | 12 | exlimiv 1489 | . . . . 5 |
14 | 2, 13 | sylbir 125 | . . . 4 |
15 | 1, 14 | sylanb 268 | . . 3 |
16 | 15 | expcom 109 | . 2 |
17 | fnrnfv 5220 | . . . 4 | |
18 | 17 | eleq2d 2107 | . . 3 |
19 | eqeq1 2046 | . . . . . 6 | |
20 | eqcom 2042 | . . . . . 6 | |
21 | 19, 20 | syl6bb 185 | . . . . 5 |
22 | 21 | rexbidv 2327 | . . . 4 |
23 | 22 | elab3g 2693 | . . 3 |
24 | 18, 23 | sylan9bbr 436 | . 2 |
25 | 16, 24 | mpancom 399 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wceq 1243 wex 1381 wcel 1393 cab 2026 wrex 2307 cvv 2557 crn 4346 wfn 4897 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: chfnrn 5278 rexrn 5304 ralrn 5305 elrnrexdmb 5307 ffnfv 5323 fconstfvm 5379 elunirn 5405 isoini 5457 reldm 5812 ordiso2 6357 uzn0 8488 frec2uzrand 9191 frecuzrdgfn 9198 uzin2 9586 |
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