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| Mirrors > Home > NFE Home > Th. List > fvmptnf | 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 5724 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 |
    |
| fvmptf.2 |
 |
| fvmptf.3 |
     |
| fvmptf.4 |
   |
| Ref | Expression |
|---|---|
| fvmptnf |
    |
,() () ()
()| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvmptf.4 |
. . . . 5
| |
| 2 | 1 | dmmptss 5685 |
. . . 4
  |
| 3 | 2 | sseli 3269 |
. . 3
|
| 4 | eqid 2353 |
. . . . . . 7
| |
| 5 | 1, 4 | fvmptex 5721 |
. . . . . 6
|
| 6 | fvex 5339 |
. . . . . . 7
| |
| 7 | fvmptf.1 |
. . . . . . . 8
| |
| 8 | nfcv 2489 |
. . . . . . . . 9
| |
| 9 | fvmptf.2 |
. . . . . . . . 9
| |
| 10 | 8, 9 | nffv 5334 |
. . . . . . . 8
|
| 11 | fvmptf.3 |
. . . . . . . . 9
| |
| 12 | 11 | fveq2d 5332 |
. . . . . . . 8
|
| 13 | 7, 10, 12, 4 | fvmptf 5722 |
. . . . . . 7
![](wedge.gif "/\"){kind=small} |
| 14 | 6, 13 | mpan2 652 |
. . . . . 6
|
| 15 | 5, 14 | syl5eq 2397 |
. . . . 5
|
| 16 | fvprc 5325 |
. . . . 5
| |
| 17 | 15, 16 | sylan9eq 2405 |
. . . 4
|
| 18 | 17 | expcom 424 |
. . 3
|
| 19 | 3, 18 | syl5 28 |
. 2
|
| 20 | ndmfv 5349 |
. 2
| |
| 21 | 19, 20 | pm2.61d1 151 |
1
|
| Colors of variables: wff setvar class |
| Syntax hints: wn 3
wi 4
wceq 1642
wcel 1710
wnfc 2476
cvv 2859
c0 3550
cid 4763
cdm 4772
cfv 4781
cmpt 5651 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
| This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-csb 3137 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-fv 4795 df-mpt 5652 |
| This theorem is referenced by: fvmptn 5724 |
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