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Mirrors > Home > MPE Home > Th. List > fvimacnv | Structured version Visualization version 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 5683 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 6022 |
. . . . 5
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2 | fvex 5902 |
. . . . . . 7
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3 | opelcnvg 5036 |
. . . . . . 7
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4 | 2, 3 | mpan 681 |
. . . . . 6
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5 | 4 | adantl 472 |
. . . . 5
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6 | 1, 5 | mpbird 240 |
. . . 4
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7 | elimasng 5216 |
. . . . . 6
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8 | 2, 7 | mpan 681 |
. . . . 5
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9 | 8 | adantl 472 |
. . . 4
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10 | 6, 9 | mpbird 240 |
. . 3
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11 | 2 | snss 4109 |
. . . . 5
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12 | imass2 5226 |
. . . . 5
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13 | 11, 12 | sylbi 200 |
. . . 4
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14 | 13 | sseld 3443 |
. . 3
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15 | 10, 14 | syl5com 31 |
. 2
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16 | fvimacnvi 6024 |
. . . 4
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17 | 16 | ex 440 |
. . 3
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18 | 17 | adantr 471 |
. 2
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19 | 15, 18 | impbid 195 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4541 ax-nul 4550 ax-pr 4656 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-rab 2758 df-v 3059 df-sbc 3280 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-br 4419 df-opab 4478 df-id 4771 df-xp 4862 df-rel 4863 df-cnv 4864 df-co 4865 df-dm 4866 df-rn 4867 df-res 4868 df-ima 4869 df-iota 5569 df-fun 5607 df-fn 5608 df-fv 5613 |
This theorem is referenced by: funimass3 6026 elpreima 6030 iinpreima 6038 isr0 20807 rnelfmlem 21022 rnelfm 21023 fmfnfmlem2 21025 fmfnfmlem4 21027 fmfnfm 21028 metustid 21624 metustsym 21625 metustexhalf 21626 xppreima 28301 dstfrvel 29356 ballotlemrv 29402 ballotlemrvOLD 29440 grpokerinj 32229 diaintclN 34672 dibintclN 34781 dihintcl 34958 arearect 36146 areaquad 36147 |
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