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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 2420 | . . . 4 | |
2 | 19.41v 1874 | . . . . 5 | |
3 | simpl 108 | . . . . . . . . . 10 | |
4 | 3 | anim1i 338 | . . . . . . . . 9 |
5 | 4 | ancomd 265 | . . . . . . . 8 |
6 | funfvex 5431 | . . . . . . . . 9 | |
7 | 6 | funfni 5218 | . . . . . . . 8 |
8 | 5, 7 | syl 14 | . . . . . . 7 |
9 | simpr 109 | . . . . . . . . 9 | |
10 | 9 | eleq1d 2206 | . . . . . . . 8 |
11 | 10 | adantr 274 | . . . . . . 7 |
12 | 8, 11 | mpbid 146 | . . . . . 6 |
13 | 12 | exlimiv 1577 | . . . . 5 |
14 | 2, 13 | sylbir 134 | . . . 4 |
15 | 1, 14 | sylanb 282 | . . 3 |
16 | 15 | expcom 115 | . 2 |
17 | fnrnfv 5461 | . . . 4 | |
18 | 17 | eleq2d 2207 | . . 3 |
19 | eqeq1 2144 | . . . . . 6 | |
20 | eqcom 2139 | . . . . . 6 | |
21 | 19, 20 | syl6bb 195 | . . . . 5 |
22 | 21 | rexbidv 2436 | . . . 4 |
23 | 22 | elab3g 2830 | . . 3 |
24 | 18, 23 | sylan9bbr 458 | . 2 |
25 | 16, 24 | mpancom 418 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wex 1468 wcel 1480 cab 2123 wrex 2415 cvv 2681 crn 4535 wfn 5113 cfv 5118 |
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-v 2683 df-sbc 2905 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-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-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 |
This theorem is referenced by: chfnrn 5524 rexrn 5550 ralrn 5551 elrnrexdmb 5553 ffnfv 5571 fconstfvm 5631 elunirn 5660 isoini 5712 reldm 6077 ordiso2 6913 eldju 6946 ctssdc 6991 uzn0 9334 frec2uzrand 10171 frecuzrdgtcl 10178 frecuzrdgfunlem 10185 uzin2 10752 |
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