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Mirrors > Home > ILE Home > Th. List > fvimacnv | Unicode version |
Description: The argument of a function value belongs to the preimage of any class containing the function value. Raph Levien remarks: "This proof is unsatisfying, because it seems to me that funimass2 4977 could probably be strengthened to a biconditional." (Contributed by Raph Levien, 20-Nov-2006.) |
Ref | Expression |
---|---|
fvimacnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfvop 5279 | . . . . 5 | |
2 | funfvex 5192 | . . . . . 6 | |
3 | opelcnvg 4515 | . . . . . 6 | |
4 | 2, 3 | sylancom 397 | . . . . 5 |
5 | 1, 4 | mpbird 156 | . . . 4 |
6 | elimasng 4693 | . . . . 5 | |
7 | 2, 6 | sylancom 397 | . . . 4 |
8 | 5, 7 | mpbird 156 | . . 3 |
9 | snssg 3500 | . . . . . . . 8 | |
10 | 2, 9 | syl 14 | . . . . . . 7 |
11 | imass2 4701 | . . . . . . 7 | |
12 | 10, 11 | syl6bi 152 | . . . . . 6 |
13 | 12 | imp 115 | . . . . 5 |
14 | 13 | sseld 2944 | . . . 4 |
15 | 14 | ex 108 | . . 3 |
16 | 8, 15 | mpid 37 | . 2 |
17 | fvimacnvi 5281 | . . . 4 | |
18 | 17 | ex 108 | . . 3 |
19 | 18 | adantr 261 | . 2 |
20 | 16, 19 | impbid 120 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 97 wb 98 wcel 1393 cvv 2557 wss 2917 csn 3375 cop 3378 ccnv 4344 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 elpreima 5286 |
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