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Mirrors > Home > ILE Home > Th. List > fnex | Unicode version |
Description: If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 5758. (Contributed by NM, 14-Aug-1994.) (Proof shortened by Andrew Salmon, 17-Sep-2011.) |
Ref | Expression |
---|---|
fnex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 5333 |
. . 3
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2 | 1 | adantr 276 |
. 2
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3 | df-fn 5238 |
. . 3
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4 | eleq1a 2261 |
. . . . . 6
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5 | 4 | impcom 125 |
. . . . 5
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6 | resfunexg 5758 |
. . . . 5
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7 | 5, 6 | sylan2 286 |
. . . 4
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8 | 7 | anassrs 400 |
. . 3
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9 | 3, 8 | sylanb 284 |
. 2
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10 | resdm 4964 |
. . . 4
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11 | 10 | eleq1d 2258 |
. . 3
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12 | 11 | biimpa 296 |
. 2
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13 | 2, 9, 12 | syl2anc 411 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 |
This theorem is referenced by: funex 5760 fex 5766 offval 6115 ofrfval 6116 tfrlemibex 6355 tfr1onlembex 6371 fndmeng 6837 cc2lem 7296 frecfzennn 10459 xpscf 12826 mulgval 13079 mulgfng 13081 invrfvald 13489 |
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