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Mirrors > Home > ILE Home > Th. List > ov2gf | Unicode version |
Description: The value of an operation class abstraction. A version of ovmpog 5905 using bound-variable hypotheses. (Contributed by NM, 17-Aug-2006.) (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
ov2gf.a | |
ov2gf.c | |
ov2gf.d | |
ov2gf.1 | |
ov2gf.2 | |
ov2gf.3 | |
ov2gf.4 | |
ov2gf.5 |
Ref | Expression |
---|---|
ov2gf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2697 | . . 3 | |
2 | ov2gf.a | . . . 4 | |
3 | ov2gf.c | . . . 4 | |
4 | ov2gf.d | . . . 4 | |
5 | ov2gf.1 | . . . . . 6 | |
6 | 5 | nfel1 2292 | . . . . 5 |
7 | ov2gf.5 | . . . . . . . 8 | |
8 | nfmpo1 5838 | . . . . . . . 8 | |
9 | 7, 8 | nfcxfr 2278 | . . . . . . 7 |
10 | nfcv 2281 | . . . . . . 7 | |
11 | 2, 9, 10 | nfov 5801 | . . . . . 6 |
12 | 11, 5 | nfeq 2289 | . . . . 5 |
13 | 6, 12 | nfim 1551 | . . . 4 |
14 | ov2gf.2 | . . . . . 6 | |
15 | 14 | nfel1 2292 | . . . . 5 |
16 | nfmpo2 5839 | . . . . . . . 8 | |
17 | 7, 16 | nfcxfr 2278 | . . . . . . 7 |
18 | 3, 17, 4 | nfov 5801 | . . . . . 6 |
19 | 18, 14 | nfeq 2289 | . . . . 5 |
20 | 15, 19 | nfim 1551 | . . . 4 |
21 | ov2gf.3 | . . . . . 6 | |
22 | 21 | eleq1d 2208 | . . . . 5 |
23 | oveq1 5781 | . . . . . 6 | |
24 | 23, 21 | eqeq12d 2154 | . . . . 5 |
25 | 22, 24 | imbi12d 233 | . . . 4 |
26 | ov2gf.4 | . . . . . 6 | |
27 | 26 | eleq1d 2208 | . . . . 5 |
28 | oveq2 5782 | . . . . . 6 | |
29 | 28, 26 | eqeq12d 2154 | . . . . 5 |
30 | 27, 29 | imbi12d 233 | . . . 4 |
31 | 7 | ovmpt4g 5893 | . . . . 5 |
32 | 31 | 3expia 1183 | . . . 4 |
33 | 2, 3, 4, 13, 20, 25, 30, 32 | vtocl2gaf 2753 | . . 3 |
34 | 1, 33 | syl5 32 | . 2 |
35 | 34 | 3impia 1178 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wceq 1331 wcel 1480 wnfc 2268 cvv 2686 (class class class)co 5774 cmpo 5776 |
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-in1 603 ax-in2 604 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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 |
This theorem is referenced by: (None) |
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