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Mirrors > Home > ILE Home > Th. List > fnexALT | 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 funimaexg 5315. This version of fnex 5754 uses ax-pow 4189 and ax-un 4448, whereas fnex 5754 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fnexALT |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 5329 |
. . . 4
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2 | relssdmrn 5164 |
. . . 4
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3 | 1, 2 | syl 14 |
. . 3
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4 | 3 | adantr 276 |
. 2
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5 | fndm 5330 |
. . . . 5
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6 | 5 | eleq1d 2258 |
. . . 4
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7 | 6 | biimpar 297 |
. . 3
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8 | fnfun 5328 |
. . . . 5
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9 | funimaexg 5315 |
. . . . 5
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10 | 8, 9 | sylan 283 |
. . . 4
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11 | imadmrn 4995 |
. . . . . . 7
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12 | 5 | imaeq2d 4985 |
. . . . . . 7
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13 | 11, 12 | eqtr3id 2236 |
. . . . . 6
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14 | 13 | eleq1d 2258 |
. . . . 5
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15 | 14 | biimpar 297 |
. . . 4
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16 | 10, 15 | syldan 282 |
. . 3
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17 | xpexg 4755 |
. . 3
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18 | 7, 16, 17 | syl2anc 411 |
. 2
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19 | ssexg 4157 |
. 2
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20 | 4, 18, 19 | 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-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 |
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-v 2754 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-br 4019 df-opab 4080 df-id 4308 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-fun 5233 df-fn 5234 |
This theorem is referenced by: (None) |
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