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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2423 |
. . . 4
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2 | 19.41v 1875 |
. . . . 5
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3 | simpl 108 |
. . . . . . . . . 10
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4 | 3 | anim1i 338 |
. . . . . . . . 9
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5 | 4 | ancomd 265 |
. . . . . . . 8
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6 | funfvex 5446 |
. . . . . . . . 9
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7 | 6 | funfni 5231 |
. . . . . . . 8
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8 | 5, 7 | syl 14 |
. . . . . . 7
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9 | simpr 109 |
. . . . . . . . 9
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10 | 9 | eleq1d 2209 |
. . . . . . . 8
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11 | 10 | adantr 274 |
. . . . . . 7
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12 | 8, 11 | mpbid 146 |
. . . . . 6
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13 | 12 | exlimiv 1578 |
. . . . 5
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14 | 2, 13 | sylbir 134 |
. . . 4
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15 | 1, 14 | sylanb 282 |
. . 3
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16 | 15 | expcom 115 |
. 2
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17 | fnrnfv 5476 |
. . . 4
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18 | 17 | eleq2d 2210 |
. . 3
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19 | eqeq1 2147 |
. . . . . 6
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20 | eqcom 2142 |
. . . . . 6
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21 | 19, 20 | syl6bb 195 |
. . . . 5
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22 | 21 | rexbidv 2439 |
. . . 4
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23 | 22 | elab3g 2839 |
. . 3
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24 | 18, 23 | sylan9bbr 459 |
. 2
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25 | 16, 24 | mpancom 419 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-fv 5139 |
This theorem is referenced by: chfnrn 5539 rexrn 5565 ralrn 5566 elrnrexdmb 5568 ffnfv 5586 fconstfvm 5646 elunirn 5675 isoini 5727 reldm 6092 ordiso2 6928 eldju 6961 ctssdc 7006 uzn0 9365 frec2uzrand 10209 frecuzrdgtcl 10216 frecuzrdgfunlem 10223 uzin2 10791 reeff1o 12902 |
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