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Mirrors > Home > MPE Home > Th. List > fvmptnf | Structured version Visualization version Unicode version |
Description: The value of a function given by an ordered-pair class abstraction is the empty set when the class it would otherwise map to is a proper class. This version of fvmptn 5983 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 21-Oct-2003.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
fvmptf.1 |
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fvmptf.2 |
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fvmptf.3 |
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fvmptf.4 |
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Ref | Expression |
---|---|
fvmptnf |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptf.4 |
. . . . 5
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2 | 1 | dmmptss 5338 |
. . . 4
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3 | 2 | sseli 3414 |
. . 3
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4 | eqid 2471 |
. . . . . . 7
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5 | 1, 4 | fvmptex 5975 |
. . . . . 6
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6 | fvex 5889 |
. . . . . . 7
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7 | fvmptf.1 |
. . . . . . . 8
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8 | nfcv 2612 |
. . . . . . . . 9
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9 | fvmptf.2 |
. . . . . . . . 9
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10 | 8, 9 | nffv 5886 |
. . . . . . . 8
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11 | fvmptf.3 |
. . . . . . . . 9
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12 | 11 | fveq2d 5883 |
. . . . . . . 8
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13 | 7, 10, 12, 4 | fvmptf 5981 |
. . . . . . 7
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14 | 6, 13 | mpan2 685 |
. . . . . 6
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15 | 5, 14 | syl5eq 2517 |
. . . . 5
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16 | fvprc 5873 |
. . . . 5
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17 | 15, 16 | sylan9eq 2525 |
. . . 4
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18 | 17 | expcom 442 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
19 | 3, 18 | syl5 32 |
. 2
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20 | ndmfv 5903 |
. 2
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21 | 19, 20 | pm2.61d1 164 |
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 1677 ax-4 1690 ax-5 1766 ax-6 1813 ax-7 1859 ax-8 1906 ax-9 1913 ax-10 1932 ax-11 1937 ax-12 1950 ax-13 2104 ax-ext 2451 ax-sep 4518 ax-nul 4527 ax-pow 4579 ax-pr 4639 |
This theorem depends on definitions: df-bi 190 df-or 377 df-an 378 df-3an 1009 df-tru 1455 df-ex 1672 df-nf 1676 df-sb 1806 df-eu 2323 df-mo 2324 df-clab 2458 df-cleq 2464 df-clel 2467 df-nfc 2601 df-ne 2643 df-ral 2761 df-rex 2762 df-rab 2765 df-v 3033 df-sbc 3256 df-csb 3350 df-dif 3393 df-un 3395 df-in 3397 df-ss 3404 df-nul 3723 df-if 3873 df-sn 3960 df-pr 3962 df-op 3966 df-uni 4191 df-br 4396 df-opab 4455 df-mpt 4456 df-id 4754 df-xp 4845 df-rel 4846 df-cnv 4847 df-co 4848 df-dm 4849 df-rn 4850 df-res 4851 df-ima 4852 df-iota 5553 df-fun 5591 df-fn 5592 df-fv 5597 |
This theorem is referenced by: fvmptn 5983 rdgsucmptnf 7165 frsucmptn 7174 |
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