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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 2461 |
. . . 4
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2 | 19.41v 1902 |
. . . . 5
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3 | simpl 109 |
. . . . . . . . . 10
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4 | 3 | anim1i 340 |
. . . . . . . . 9
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5 | 4 | ancomd 267 |
. . . . . . . 8
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6 | funfvex 5532 |
. . . . . . . . 9
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7 | 6 | funfni 5316 |
. . . . . . . 8
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8 | 5, 7 | syl 14 |
. . . . . . 7
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9 | simpr 110 |
. . . . . . . . 9
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10 | 9 | eleq1d 2246 |
. . . . . . . 8
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11 | 10 | adantr 276 |
. . . . . . 7
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12 | 8, 11 | mpbid 147 |
. . . . . 6
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13 | 12 | exlimiv 1598 |
. . . . 5
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14 | 2, 13 | sylbir 135 |
. . . 4
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15 | 1, 14 | sylanb 284 |
. . 3
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16 | 15 | expcom 116 |
. 2
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17 | fnrnfv 5562 |
. . . 4
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18 | 17 | eleq2d 2247 |
. . 3
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19 | eqeq1 2184 |
. . . . . 6
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20 | eqcom 2179 |
. . . . . 6
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21 | 19, 20 | bitrdi 196 |
. . . . 5
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22 | 21 | rexbidv 2478 |
. . . 4
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23 | 22 | elab3g 2888 |
. . 3
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24 | 18, 23 | sylan9bbr 463 |
. 2
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25 | 16, 24 | mpancom 422 |
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 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 |
This theorem is referenced by: foelcdmi 5568 chfnrn 5627 rexrn 5653 ralrn 5654 elrnrexdmb 5656 ffnfv 5674 fconstfvm 5734 elunirn 5766 isoini 5818 canth 5828 reldm 6186 ordiso2 7033 eldju 7066 ctssdc 7111 uzn0 9541 frec2uzrand 10402 frecuzrdgtcl 10409 frecuzrdgfunlem 10416 uzin2 10991 reeff1o 14087 |
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