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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 4986 |
. . . . 5
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2 | 1 | sseli 3151 |
. . . 4
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3 | fndm 5310 |
. . . . 5
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4 | 3 | eleq2d 2247 |
. . . 4
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5 | 2, 4 | syl5ib 154 |
. . 3
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6 | fnfun 5308 |
. . . . 5
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7 | fvimacnvi 5625 |
. . . . 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 5626 |
. . . . 5
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12 | 11 | funfni 5311 |
. . . 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 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 4118 ax-pow 4171 ax-pr 4205 |
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 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 |
This theorem is referenced by: fniniseg 5631 fncnvima2 5632 rexsupp 5635 unpreima 5636 respreima 5639 fisumss 11371 fprodssdc 11569 tanvalap 11687 1arith 12335 cncnpi 13361 cncnp 13363 cnpdis 13375 tx1cn 13402 tx2cn 13403 txcnmpt 13406 txdis1cn 13411 xmeterval 13568 cnbl0 13667 cnblcld 13668 |
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