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Mirrors > Home > ILE Home > Th. List > fvimacnv | Unicode version |
Description: The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 5209 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.) |
Ref | Expression |
---|---|
fvimacnv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfvop 5540 |
. . . . 5
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2 | funfvex 5446 |
. . . . . 6
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3 | opelcnvg 4727 |
. . . . . 6
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4 | 2, 3 | sylancom 417 |
. . . . 5
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5 | 1, 4 | mpbird 166 |
. . . 4
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6 | elimasng 4915 |
. . . . 5
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7 | 2, 6 | sylancom 417 |
. . . 4
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8 | 5, 7 | mpbird 166 |
. . 3
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9 | snssg 3664 |
. . . . . . . 8
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10 | 2, 9 | syl 14 |
. . . . . . 7
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11 | imass2 4923 |
. . . . . . 7
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12 | 10, 11 | syl6bi 162 |
. . . . . 6
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13 | 12 | imp 123 |
. . . . 5
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14 | 13 | sseld 3101 |
. . . 4
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15 | 14 | ex 114 |
. . 3
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16 | 8, 15 | mpid 42 |
. 2
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17 | fvimacnvi 5542 |
. . . 4
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18 | 17 | ex 114 |
. . 3
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19 | 18 | adantr 274 |
. 2
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20 | 16, 19 | impbid 128 |
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 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-sbc 2914 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-fv 5139 |
This theorem is referenced by: funimass3 5544 elpreima 5547 fisumss 11193 |
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