Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2422 | . . . 4 | |
2 | 19.41v 1874 | . . . . 5 | |
3 | simpl 108 | . . . . . . . . . 10 | |
4 | 3 | anim1i 338 | . . . . . . . . 9 |
5 | 4 | ancomd 265 | . . . . . . . 8 |
6 | funfvex 5438 | . . . . . . . . 9 | |
7 | 6 | funfni 5223 | . . . . . . . 8 |
8 | 5, 7 | syl 14 | . . . . . . 7 |
9 | simpr 109 | . . . . . . . . 9 | |
10 | 9 | eleq1d 2208 | . . . . . . . 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 5468 | . . . 4 | |
18 | 17 | eleq2d 2209 | . . 3 |
19 | eqeq1 2146 | . . . . . 6 | |
20 | eqcom 2141 | . . . . . 6 | |
21 | 19, 20 | syl6bb 195 | . . . . 5 |
22 | 21 | rexbidv 2438 | . . . 4 |
23 | 22 | elab3g 2835 | . . 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 2125 wrex 2417 cvv 2686 crn 4540 wfn 5118 cfv 5123 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: chfnrn 5531 rexrn 5557 ralrn 5558 elrnrexdmb 5560 ffnfv 5578 fconstfvm 5638 elunirn 5667 isoini 5719 reldm 6084 ordiso2 6920 eldju 6953 ctssdc 6998 uzn0 9341 frec2uzrand 10178 frecuzrdgtcl 10185 frecuzrdgfunlem 10192 uzin2 10759 |
Copyright terms: Public domain | W3C validator |