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Mirrors > Home > ILE Home > Th. List > funimass3 | Unicode version |
Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5583 would be the special case of being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.) |
Ref | Expression |
---|---|
funimass3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funimass4 5520 | . . 3 | |
2 | ssel 3122 | . . . . . 6 | |
3 | fvimacnv 5583 | . . . . . . 7 | |
4 | 3 | ex 114 | . . . . . 6 |
5 | 2, 4 | syl9r 73 | . . . . 5 |
6 | 5 | imp31 254 | . . . 4 |
7 | 6 | ralbidva 2453 | . . 3 |
8 | 1, 7 | bitrd 187 | . 2 |
9 | dfss3 3118 | . 2 | |
10 | 8, 9 | bitr4di 197 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wcel 2128 wral 2435 wss 3102 ccnv 4586 cdm 4587 cima 4590 wfun 5165 cfv 5171 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-fv 5179 |
This theorem is referenced by: funimass5 5585 funconstss 5586 fimacnv 5597 iscnp3 12645 cnpnei 12661 cncnp 12672 |
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