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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 3142 | . 2 | |
2 | vex 2738 | . . . . . . . . 9 | |
3 | 2 | elima 4968 | . . . . . . . 8 |
4 | eqcom 2177 | . . . . . . . . . 10 | |
5 | ssel 3147 | . . . . . . . . . . . 12 | |
6 | funbrfvb 5550 | . . . . . . . . . . . . 13 | |
7 | 6 | ex 115 | . . . . . . . . . . . 12 |
8 | 5, 7 | syl9 72 | . . . . . . . . . . 11 |
9 | 8 | imp31 256 | . . . . . . . . . 10 |
10 | 4, 9 | bitrid 192 | . . . . . . . . 9 |
11 | 10 | rexbidva 2472 | . . . . . . . 8 |
12 | 3, 11 | bitr4id 199 | . . . . . . 7 |
13 | 12 | imbi1d 231 | . . . . . 6 |
14 | r19.23v 2584 | . . . . . 6 | |
15 | 13, 14 | bitr4di 198 | . . . . 5 |
16 | 15 | albidv 1822 | . . . 4 |
17 | 16 | ancoms 268 | . . 3 |
18 | ralcom4 2757 | . . . 4 | |
19 | ssel2 3148 | . . . . . . . . 9 | |
20 | 19 | anim2i 342 | . . . . . . . 8 |
21 | 20 | 3impb 1199 | . . . . . . 7 |
22 | funfvex 5524 | . . . . . . 7 | |
23 | nfv 1526 | . . . . . . . 8 | |
24 | eleq1 2238 | . . . . . . . 8 | |
25 | 23, 24 | ceqsalg 2763 | . . . . . . 7 |
26 | 21, 22, 25 | 3syl 17 | . . . . . 6 |
27 | 26 | 3expa 1203 | . . . . 5 |
28 | 27 | ralbidva 2471 | . . . 4 |
29 | 18, 28 | bitr3id 194 | . . 3 |
30 | 17, 29 | bitrd 188 | . 2 |
31 | 1, 30 | bitrid 192 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 104 wb 105 w3a 978 wal 1351 wceq 1353 wcel 2146 wral 2453 wrex 2454 cvv 2735 wss 3127 class class class wbr 3998 cdm 4620 cima 4623 wfun 5202 cfv 5208 |
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 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-sbc 2961 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-br 3999 df-opab 4060 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-fv 5216 |
This theorem is referenced by: funimass3 5624 funimass5 5625 funconstss 5626 funimassov 6014 phimullem 12191 txcnp 13264 metcnp 13505 |
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