| Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > ILE Home > Th. List > fvmptss2 | Unicode version | ||
| Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.) |
| Ref | Expression |
|---|---|
| fvmptss2.1 |
|
| fvmptss2.2 |
|
| Ref | Expression |
|---|---|
| fvmptss2 |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fvss 5589 |
. 2
| |
| 2 | fvmptss2.2 |
. . . . . 6
| |
| 3 | 2 | funmpt2 5309 |
. . . . 5
|
| 4 | funrel 5287 |
. . . . 5
| |
| 5 | 3, 4 | ax-mp 5 |
. . . 4
|
| 6 | 5 | brrelex1i 4717 |
. . 3
|
| 7 | nfcv 2347 |
. . . 4
| |
| 8 | nfmpt1 4136 |
. . . . . . 7
| |
| 9 | 2, 8 | nfcxfr 2344 |
. . . . . 6
|
| 10 | nfcv 2347 |
. . . . . 6
| |
| 11 | 7, 9, 10 | nfbr 4089 |
. . . . 5
|
| 12 | nfv 1550 |
. . . . 5
| |
| 13 | 11, 12 | nfim 1594 |
. . . 4
|
| 14 | breq1 4046 |
. . . . 5
| |
| 15 | fvmptss2.1 |
. . . . . 6
| |
| 16 | 15 | sseq2d 3222 |
. . . . 5
|
| 17 | 14, 16 | imbi12d 234 |
. . . 4
|
| 18 | df-br 4044 |
. . . . 5
| |
| 19 | opabid 4301 |
. . . . . . 7
| |
| 20 | eqimss 3246 |
. . . . . . . 8
| |
| 21 | 20 | adantl 277 |
. . . . . . 7
|
| 22 | 19, 21 | sylbi 121 |
. . . . . 6
|
| 23 | df-mpt 4106 |
. . . . . . 7
| |
| 24 | 2, 23 | eqtri 2225 |
. . . . . 6
|
| 25 | 22, 24 | eleq2s 2299 |
. . . . 5
|
| 26 | 18, 25 | sylbi 121 |
. . . 4
|
| 27 | 7, 13, 17, 26 | vtoclgf 2830 |
. . 3
|
| 28 | 6, 27 | mpcom 36 |
. 2
|
| 29 | 1, 28 | mpg 1473 |
1
|
| 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 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-v 2773 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4339 df-xp 4680 df-rel 4681 df-cnv 4682 df-co 4683 df-iota 5231 df-fun 5272 df-fv 5278 |
| This theorem is referenced by: mptfvex 5664 |
| Copyright terms: Public domain | W3C validator |