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Mirrors > Home > ILE Home > Th. List > ovmpodf | Unicode version |
Description: Alternate deduction version of ovmpo 5906, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
Ref | Expression |
---|---|
ovmpodf.1 | |
ovmpodf.2 | |
ovmpodf.3 | |
ovmpodf.4 | |
ovmpodf.5 | |
ovmpodf.6 | |
ovmpodf.7 | |
ovmpodf.8 |
Ref | Expression |
---|---|
ovmpodf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1508 | . 2 | |
2 | ovmpodf.5 | . . . 4 | |
3 | nfmpo1 5838 | . . . 4 | |
4 | 2, 3 | nfeq 2289 | . . 3 |
5 | ovmpodf.6 | . . 3 | |
6 | 4, 5 | nfim 1551 | . 2 |
7 | ovmpodf.1 | . . . 4 | |
8 | elex 2697 | . . . 4 | |
9 | 7, 8 | syl 14 | . . 3 |
10 | isset 2692 | . . 3 | |
11 | 9, 10 | sylib 121 | . 2 |
12 | ovmpodf.2 | . . . . 5 | |
13 | elex 2697 | . . . . 5 | |
14 | 12, 13 | syl 14 | . . . 4 |
15 | isset 2692 | . . . 4 | |
16 | 14, 15 | sylib 121 | . . 3 |
17 | nfv 1508 | . . . 4 | |
18 | ovmpodf.7 | . . . . . 6 | |
19 | nfmpo2 5839 | . . . . . 6 | |
20 | 18, 19 | nfeq 2289 | . . . . 5 |
21 | ovmpodf.8 | . . . . 5 | |
22 | 20, 21 | nfim 1551 | . . . 4 |
23 | oveq 5780 | . . . . . 6 | |
24 | simprl 520 | . . . . . . . . . 10 | |
25 | simprr 521 | . . . . . . . . . 10 | |
26 | 24, 25 | oveq12d 5792 | . . . . . . . . 9 |
27 | 7 | adantr 274 | . . . . . . . . . . 11 |
28 | 24, 27 | eqeltrd 2216 | . . . . . . . . . 10 |
29 | 12 | adantrr 470 | . . . . . . . . . . 11 |
30 | 25, 29 | eqeltrd 2216 | . . . . . . . . . 10 |
31 | ovmpodf.3 | . . . . . . . . . 10 | |
32 | eqid 2139 | . . . . . . . . . . 11 | |
33 | 32 | ovmpt4g 5893 | . . . . . . . . . 10 |
34 | 28, 30, 31, 33 | syl3anc 1216 | . . . . . . . . 9 |
35 | 26, 34 | eqtr3d 2174 | . . . . . . . 8 |
36 | 35 | eqeq2d 2151 | . . . . . . 7 |
37 | ovmpodf.4 | . . . . . . 7 | |
38 | 36, 37 | sylbid 149 | . . . . . 6 |
39 | 23, 38 | syl5 32 | . . . . 5 |
40 | 39 | expr 372 | . . . 4 |
41 | 17, 22, 40 | exlimd 1576 | . . 3 |
42 | 16, 41 | mpd 13 | . 2 |
43 | 1, 6, 11, 42 | exlimdd 1844 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wnf 1436 wex 1468 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: ovmpodv 5903 ovmpodv2 5904 |
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