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Mirrors > Home > ILE Home > Th. List > ovmpos | Unicode version |
Description: Value of a function given by the maps-to notation, expressed using explicit substitution. (Contributed by Mario Carneiro, 30-Apr-2015.) |
Ref | Expression |
---|---|
ovmpos.3 |
Ref | Expression |
---|---|
ovmpos |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2741 | . . 3 | |
2 | nfcv 2312 | . . . . 5 | |
3 | nfcv 2312 | . . . . 5 | |
4 | nfcv 2312 | . . . . 5 | |
5 | nfcsb1v 3082 | . . . . . . 7 | |
6 | 5 | nfel1 2323 | . . . . . 6 |
7 | ovmpos.3 | . . . . . . . . 9 | |
8 | nfmpo1 5920 | . . . . . . . . 9 | |
9 | 7, 8 | nfcxfr 2309 | . . . . . . . 8 |
10 | nfcv 2312 | . . . . . . . 8 | |
11 | 2, 9, 10 | nfov 5883 | . . . . . . 7 |
12 | 11, 5 | nfeq 2320 | . . . . . 6 |
13 | 6, 12 | nfim 1565 | . . . . 5 |
14 | nfcsb1v 3082 | . . . . . . 7 | |
15 | 14 | nfel1 2323 | . . . . . 6 |
16 | nfmpo2 5921 | . . . . . . . . 9 | |
17 | 7, 16 | nfcxfr 2309 | . . . . . . . 8 |
18 | 3, 17, 4 | nfov 5883 | . . . . . . 7 |
19 | 18, 14 | nfeq 2320 | . . . . . 6 |
20 | 15, 19 | nfim 1565 | . . . . 5 |
21 | csbeq1a 3058 | . . . . . . 7 | |
22 | 21 | eleq1d 2239 | . . . . . 6 |
23 | oveq1 5860 | . . . . . . 7 | |
24 | 23, 21 | eqeq12d 2185 | . . . . . 6 |
25 | 22, 24 | imbi12d 233 | . . . . 5 |
26 | csbeq1a 3058 | . . . . . . 7 | |
27 | 26 | eleq1d 2239 | . . . . . 6 |
28 | oveq2 5861 | . . . . . . 7 | |
29 | 28, 26 | eqeq12d 2185 | . . . . . 6 |
30 | 27, 29 | imbi12d 233 | . . . . 5 |
31 | 7 | ovmpt4g 5975 | . . . . . 6 |
32 | 31 | 3expia 1200 | . . . . 5 |
33 | 2, 3, 4, 13, 20, 25, 30, 32 | vtocl2gaf 2797 | . . . 4 |
34 | csbcomg 3072 | . . . . 5 | |
35 | 34 | eleq1d 2239 | . . . 4 |
36 | 34 | eqeq2d 2182 | . . . 4 |
37 | 33, 35, 36 | 3imtr4d 202 | . . 3 |
38 | 1, 37 | syl5 32 | . 2 |
39 | 38 | 3impia 1195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wcel 2141 cvv 2730 csb 3049 (class class class)co 5853 cmpo 5855 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 |
This theorem is referenced by: (None) |
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