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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 |
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fvmptss2.2 |
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Ref | Expression |
---|---|
fvmptss2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvss 5332 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | fvmptss2.2 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
3 | 2 | funmpt2 5066 |
. . . . 5
![]() ![]() ![]() |
4 | funrel 5045 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | 3, 4 | ax-mp 7 |
. . . 4
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6 | 5 | brrelex1i 4494 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7 | nfcv 2229 |
. . . 4
![]() ![]() ![]() ![]() | |
8 | nfmpt1 3937 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
9 | 2, 8 | nfcxfr 2226 |
. . . . . 6
![]() ![]() ![]() ![]() |
10 | nfcv 2229 |
. . . . . 6
![]() ![]() ![]() ![]() | |
11 | 7, 9, 10 | nfbr 3895 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() |
12 | nfv 1467 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() | |
13 | 11, 12 | nfim 1510 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | breq1 3854 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | fvmptss2.1 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
16 | 15 | sseq2d 3055 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
17 | 14, 16 | imbi12d 233 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
18 | df-br 3852 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | opabid 4093 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
20 | eqimss 3079 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 20 | adantl 272 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 19, 21 | sylbi 120 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
23 | df-mpt 3907 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
24 | 2, 23 | eqtri 2109 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
25 | 22, 24 | eleq2s 2183 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
26 | 18, 25 | sylbi 120 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 | 7, 13, 17, 26 | vtoclgf 2678 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
28 | 6, 27 | mpcom 36 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | 1, 28 | mpg 1386 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ral 2365 df-rex 2366 df-v 2622 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-mpt 3907 df-id 4129 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-iota 4993 df-fun 5030 df-fv 5036 |
This theorem is referenced by: mptfvex 5401 |
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