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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 3126 | . 2 | |
2 | vex 2724 | . . . . . . . . 9 | |
3 | 2 | elima 4945 | . . . . . . . 8 |
4 | eqcom 2166 | . . . . . . . . . 10 | |
5 | ssel 3131 | . . . . . . . . . . . 12 | |
6 | funbrfvb 5523 | . . . . . . . . . . . . 13 | |
7 | 6 | ex 114 | . . . . . . . . . . . 12 |
8 | 5, 7 | syl9 72 | . . . . . . . . . . 11 |
9 | 8 | imp31 254 | . . . . . . . . . 10 |
10 | 4, 9 | syl5bb 191 | . . . . . . . . 9 |
11 | 10 | rexbidva 2461 | . . . . . . . 8 |
12 | 3, 11 | bitr4id 198 | . . . . . . 7 |
13 | 12 | imbi1d 230 | . . . . . 6 |
14 | r19.23v 2573 | . . . . . 6 | |
15 | 13, 14 | bitr4di 197 | . . . . 5 |
16 | 15 | albidv 1811 | . . . 4 |
17 | 16 | ancoms 266 | . . 3 |
18 | ralcom4 2743 | . . . 4 | |
19 | ssel2 3132 | . . . . . . . . 9 | |
20 | 19 | anim2i 340 | . . . . . . . 8 |
21 | 20 | 3impb 1188 | . . . . . . 7 |
22 | funfvex 5497 | . . . . . . 7 | |
23 | nfv 1515 | . . . . . . . 8 | |
24 | eleq1 2227 | . . . . . . . 8 | |
25 | 23, 24 | ceqsalg 2749 | . . . . . . 7 |
26 | 21, 22, 25 | 3syl 17 | . . . . . 6 |
27 | 26 | 3expa 1192 | . . . . 5 |
28 | 27 | ralbidva 2460 | . . . 4 |
29 | 18, 28 | bitr3id 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 967 wal 1340 wceq 1342 wcel 2135 wral 2442 wrex 2443 cvv 2721 wss 3111 class class class wbr 3976 cdm 4598 cima 4601 wfun 5176 cfv 5182 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 |
This theorem is referenced by: funimass3 5595 funimass5 5596 funconstss 5597 funimassov 5982 phimullem 12134 txcnp 12812 metcnp 13053 |
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