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Mirrors > Home > ILE Home > Th. List > funimass4 | Unicode version |
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.) |
Ref | Expression |
---|---|
funimass4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 2934 | . 2 | |
2 | eqcom 2042 | . . . . . . . . . 10 | |
3 | ssel 2939 | . . . . . . . . . . . 12 | |
4 | funbrfvb 5216 | . . . . . . . . . . . . 13 | |
5 | 4 | ex 108 | . . . . . . . . . . . 12 |
6 | 3, 5 | syl9 66 | . . . . . . . . . . 11 |
7 | 6 | imp31 243 | . . . . . . . . . 10 |
8 | 2, 7 | syl5bb 181 | . . . . . . . . 9 |
9 | 8 | rexbidva 2323 | . . . . . . . 8 |
10 | vex 2560 | . . . . . . . . 9 | |
11 | 10 | elima 4673 | . . . . . . . 8 |
12 | 9, 11 | syl6rbbr 188 | . . . . . . 7 |
13 | 12 | imbi1d 220 | . . . . . 6 |
14 | r19.23v 2425 | . . . . . 6 | |
15 | 13, 14 | syl6bbr 187 | . . . . 5 |
16 | 15 | albidv 1705 | . . . 4 |
17 | 16 | ancoms 255 | . . 3 |
18 | ralcom4 2576 | . . . 4 | |
19 | ssel2 2940 | . . . . . . . . 9 | |
20 | 19 | anim2i 324 | . . . . . . . 8 |
21 | 20 | 3impb 1100 | . . . . . . 7 |
22 | funfvex 5192 | . . . . . . 7 | |
23 | nfv 1421 | . . . . . . . 8 | |
24 | eleq1 2100 | . . . . . . . 8 | |
25 | 23, 24 | ceqsalg 2582 | . . . . . . 7 |
26 | 21, 22, 25 | 3syl 17 | . . . . . 6 |
27 | 26 | 3expa 1104 | . . . . 5 |
28 | 27 | ralbidva 2322 | . . . 4 |
29 | 18, 28 | syl5bbr 183 | . . 3 |
30 | 17, 29 | bitrd 177 | . 2 |
31 | 1, 30 | syl5bb 181 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 w3a 885 wal 1241 wceq 1243 wcel 1393 wral 2306 wrex 2307 cvv 2557 wss 2917 class class class wbr 3764 cdm 4345 cima 4348 wfun 4896 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: funimass3 5283 funimass5 5284 funconstss 5285 funimassov 5650 |
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