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Mirrors > Home > ILE Home > Th. List > elpreima | Unicode version |
Description: Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
elpreima |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvimass 5009 |
. . . . 5
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2 | 1 | sseli 3166 |
. . . 4
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3 | fndm 5334 |
. . . . 5
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4 | 3 | eleq2d 2259 |
. . . 4
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5 | 2, 4 | imbitrid 154 |
. . 3
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6 | fnfun 5332 |
. . . . 5
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7 | fvimacnvi 5651 |
. . . . 5
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8 | 6, 7 | sylan 283 |
. . . 4
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9 | 8 | ex 115 |
. . 3
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10 | 5, 9 | jcad 307 |
. 2
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11 | fvimacnv 5652 |
. . . . 5
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12 | 11 | funfni 5335 |
. . . 4
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13 | 12 | biimpd 144 |
. . 3
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14 | 13 | expimpd 363 |
. 2
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15 | 10, 14 | impbid 129 |
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-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-v 2754 df-sbc 2978 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 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-fv 5243 |
This theorem is referenced by: fniniseg 5657 fncnvima2 5658 rexsupp 5661 unpreima 5662 respreima 5665 fisumss 11432 fprodssdc 11630 tanvalap 11748 1arith 12399 ghmpreima 13205 ghmnsgpreima 13208 kerf1ghm 13213 psrbaglesuppg 13950 cncnpi 14188 cncnp 14190 cnpdis 14202 tx1cn 14229 tx2cn 14230 txcnmpt 14233 txdis1cn 14238 xmeterval 14395 cnbl0 14494 cnblcld 14495 |
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