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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 4795 |
. . . . 5
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2 | 1 | sseli 3021 |
. . . 4
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3 | fndm 5113 |
. . . . 5
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4 | 3 | eleq2d 2157 |
. . . 4
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5 | 2, 4 | syl5ib 152 |
. . 3
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6 | fnfun 5111 |
. . . . 5
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7 | fvimacnvi 5413 |
. . . . 5
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8 | 6, 7 | sylan 277 |
. . . 4
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9 | 8 | ex 113 |
. . 3
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10 | 5, 9 | jcad 301 |
. 2
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11 | fvimacnv 5414 |
. . . . 5
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12 | 11 | funfni 5114 |
. . . 4
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13 | 12 | biimpd 142 |
. . 3
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14 | 13 | expimpd 355 |
. 2
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15 | 10, 14 | impbid 127 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-sbc 2841 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-rn 4449 df-res 4450 df-ima 4451 df-iota 4980 df-fun 5017 df-fn 5018 df-fv 5023 |
This theorem is referenced by: fniniseg 5419 fncnvima2 5420 rexsupp 5423 unpreima 5424 respreima 5427 fisumss 10784 tanvalap 10999 |
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