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Mirrors > Home > ILE Home > Th. List > ovmpodf | Unicode version |
Description: Alternate deduction version of ovmpo 5968, 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 1515 | . 2 | |
2 | ovmpodf.5 | . . . 4 | |
3 | nfmpo1 5900 | . . . 4 | |
4 | 2, 3 | nfeq 2314 | . . 3 |
5 | ovmpodf.6 | . . 3 | |
6 | 4, 5 | nfim 1559 | . 2 |
7 | ovmpodf.1 | . . . 4 | |
8 | elex 2732 | . . . 4 | |
9 | 7, 8 | syl 14 | . . 3 |
10 | isset 2727 | . . 3 | |
11 | 9, 10 | sylib 121 | . 2 |
12 | ovmpodf.2 | . . . . 5 | |
13 | elex 2732 | . . . . 5 | |
14 | 12, 13 | syl 14 | . . . 4 |
15 | isset 2727 | . . . 4 | |
16 | 14, 15 | sylib 121 | . . 3 |
17 | nfv 1515 | . . . 4 | |
18 | ovmpodf.7 | . . . . . 6 | |
19 | nfmpo2 5901 | . . . . . 6 | |
20 | 18, 19 | nfeq 2314 | . . . . 5 |
21 | ovmpodf.8 | . . . . 5 | |
22 | 20, 21 | nfim 1559 | . . . 4 |
23 | oveq 5842 | . . . . . 6 | |
24 | simprl 521 | . . . . . . . . . 10 | |
25 | simprr 522 | . . . . . . . . . 10 | |
26 | 24, 25 | oveq12d 5854 | . . . . . . . . 9 |
27 | 7 | adantr 274 | . . . . . . . . . . 11 |
28 | 24, 27 | eqeltrd 2241 | . . . . . . . . . 10 |
29 | 12 | adantrr 471 | . . . . . . . . . . 11 |
30 | 25, 29 | eqeltrd 2241 | . . . . . . . . . 10 |
31 | ovmpodf.3 | . . . . . . . . . 10 | |
32 | eqid 2164 | . . . . . . . . . . 11 | |
33 | 32 | ovmpt4g 5955 | . . . . . . . . . 10 |
34 | 28, 30, 31, 33 | syl3anc 1227 | . . . . . . . . 9 |
35 | 26, 34 | eqtr3d 2199 | . . . . . . . 8 |
36 | 35 | eqeq2d 2176 | . . . . . . 7 |
37 | ovmpodf.4 | . . . . . . 7 | |
38 | 36, 37 | sylbid 149 | . . . . . 6 |
39 | 23, 38 | syl5 32 | . . . . 5 |
40 | 39 | expr 373 | . . . 4 |
41 | 17, 22, 40 | exlimd 1584 | . . 3 |
42 | 16, 41 | mpd 13 | . 2 |
43 | 1, 6, 11, 42 | exlimdd 1859 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1342 wnf 1447 wex 1479 wcel 2135 wnfc 2293 cvv 2721 (class class class)co 5836 cmpo 5838 |
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 604 ax-in2 605 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 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 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-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-dif 3113 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-iota 5147 df-fun 5184 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 |
This theorem is referenced by: ovmpodv 5965 ovmpodv2 5966 |
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