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Mirrors > Home > NFE Home > Th. List > fvmptex | Unicode version |
Description: Express a function whose value may not always be a set in terms of another function for which sethood is guaranteed. (Note that is just shorthand for , and it is always a set by fvex 5339.) Note also that these functions are not the same; wherever is not a set, is not in the domain of (so it evaluates to the empty set), but is in the domain of , and is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.) |
Ref | Expression |
---|---|
fvmptex.1 | |
fvmptex.2 |
Ref | Expression |
---|---|
fvmptex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbeq1 3139 | . . . 4 | |
2 | fvmptex.1 | . . . . 5 | |
3 | nfcv 2489 | . . . . . 6 | |
4 | nfcsb1v 3168 | . . . . . 6 | |
5 | csbeq1a 3144 | . . . . . 6 | |
6 | 3, 4, 5 | cbvmpt 5676 | . . . . 5 |
7 | 2, 6 | eqtri 2373 | . . . 4 |
8 | 1, 7 | fvmpti 5699 | . . 3 |
9 | 1 | fveq2d 5332 | . . . 4 |
10 | fvmptex.2 | . . . . 5 | |
11 | nfcv 2489 | . . . . . 6 | |
12 | nfcv 2489 | . . . . . . 7 | |
13 | 12, 4 | nffv 5334 | . . . . . 6 |
14 | 5 | fveq2d 5332 | . . . . . 6 |
15 | 11, 13, 14 | cbvmpt 5676 | . . . . 5 |
16 | 10, 15 | eqtri 2373 | . . . 4 |
17 | fvex 5339 | . . . 4 | |
18 | 9, 16, 17 | fvmpt 5700 | . . 3 |
19 | 8, 18 | eqtr4d 2388 | . 2 |
20 | 2 | dmmptss 5685 | . . . . . 6 |
21 | 20 | sseli 3269 | . . . . 5 |
22 | 21 | con3i 127 | . . . 4 |
23 | ndmfv 5349 | . . . 4 | |
24 | 22, 23 | syl 15 | . . 3 |
25 | fvex 5339 | . . . . . 6 | |
26 | 25, 10 | dmmpti 5691 | . . . . 5 |
27 | 26 | eleq2i 2417 | . . . 4 |
28 | ndmfv 5349 | . . . 4 | |
29 | 27, 28 | sylnbir 298 | . . 3 |
30 | 24, 29 | eqtr4d 2388 | . 2 |
31 | 19, 30 | pm2.61i 156 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wceq 1642 wcel 1710 csb 3136 c0 3550 cid 4763 cdm 4772 cfv 4781 cmpt 5651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 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: fvmptnf 5723 |
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