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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 2448 | . . . 4 | |
2 | 19.41v 1889 | . . . . 5 | |
3 | simpl 108 | . . . . . . . . . 10 | |
4 | 3 | anim1i 338 | . . . . . . . . 9 |
5 | 4 | ancomd 265 | . . . . . . . 8 |
6 | funfvex 5497 | . . . . . . . . 9 | |
7 | 6 | funfni 5282 | . . . . . . . 8 |
8 | 5, 7 | syl 14 | . . . . . . 7 |
9 | simpr 109 | . . . . . . . . 9 | |
10 | 9 | eleq1d 2233 | . . . . . . . 8 |
11 | 10 | adantr 274 | . . . . . . 7 |
12 | 8, 11 | mpbid 146 | . . . . . 6 |
13 | 12 | exlimiv 1585 | . . . . 5 |
14 | 2, 13 | sylbir 134 | . . . 4 |
15 | 1, 14 | sylanb 282 | . . 3 |
16 | 15 | expcom 115 | . 2 |
17 | fnrnfv 5527 | . . . 4 | |
18 | 17 | eleq2d 2234 | . . 3 |
19 | eqeq1 2171 | . . . . . 6 | |
20 | eqcom 2166 | . . . . . 6 | |
21 | 19, 20 | bitrdi 195 | . . . . 5 |
22 | 21 | rexbidv 2465 | . . . 4 |
23 | 22 | elab3g 2872 | . . 3 |
24 | 18, 23 | sylan9bbr 459 | . 2 |
25 | 16, 24 | mpancom 419 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 cab 2150 wrex 2443 cvv 2721 crn 4599 wfn 5177 cfv 5182 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 |
This theorem is referenced by: chfnrn 5590 rexrn 5616 ralrn 5617 elrnrexdmb 5619 ffnfv 5637 fconstfvm 5697 elunirn 5728 isoini 5780 canth 5790 reldm 6146 ordiso2 6991 eldju 7024 ctssdc 7069 uzn0 9472 frec2uzrand 10330 frecuzrdgtcl 10337 frecuzrdgfunlem 10344 uzin2 10915 reeff1o 13241 |
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