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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 3081 | . 2 | |
2 | eqcom 2139 | . . . . . . . . . 10 | |
3 | ssel 3086 | . . . . . . . . . . . 12 | |
4 | funbrfvb 5457 | . . . . . . . . . . . . 13 | |
5 | 4 | ex 114 | . . . . . . . . . . . 12 |
6 | 3, 5 | syl9 72 | . . . . . . . . . . 11 |
7 | 6 | imp31 254 | . . . . . . . . . 10 |
8 | 2, 7 | syl5bb 191 | . . . . . . . . 9 |
9 | 8 | rexbidva 2432 | . . . . . . . 8 |
10 | vex 2684 | . . . . . . . . 9 | |
11 | 10 | elima 4881 | . . . . . . . 8 |
12 | 9, 11 | syl6rbbr 198 | . . . . . . 7 |
13 | 12 | imbi1d 230 | . . . . . 6 |
14 | r19.23v 2539 | . . . . . 6 | |
15 | 13, 14 | syl6bbr 197 | . . . . 5 |
16 | 15 | albidv 1796 | . . . 4 |
17 | 16 | ancoms 266 | . . 3 |
18 | ralcom4 2703 | . . . 4 | |
19 | ssel2 3087 | . . . . . . . . 9 | |
20 | 19 | anim2i 339 | . . . . . . . 8 |
21 | 20 | 3impb 1177 | . . . . . . 7 |
22 | funfvex 5431 | . . . . . . 7 | |
23 | nfv 1508 | . . . . . . . 8 | |
24 | eleq1 2200 | . . . . . . . 8 | |
25 | 23, 24 | ceqsalg 2709 | . . . . . . 7 |
26 | 21, 22, 25 | 3syl 17 | . . . . . 6 |
27 | 26 | 3expa 1181 | . . . . 5 |
28 | 27 | ralbidva 2431 | . . . 4 |
29 | 18, 28 | syl5bbr 193 | . . 3 |
30 | 17, 29 | bitrd 187 | . 2 |
31 | 1, 30 | syl5bb 191 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wal 1329 wceq 1331 wcel 1480 wral 2414 wrex 2415 cvv 2681 wss 3066 class class class wbr 3924 cdm 4534 cima 4537 wfun 5112 cfv 5118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 |
This theorem is referenced by: funimass3 5529 funimass5 5530 funconstss 5531 funimassov 5913 phimullem 11890 txcnp 12429 metcnp 12670 |
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