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Mirrors > Home > NFE Home > Th. List > eloprabga | Unicode version |
Description: The law of concretion for operation class abstraction. Compare elopab 4696. (Contributed by set.mm contributors, 17-Dec-2013.) (Revised by David Abernethy, 18-Jun-2012.) Removed unnecessary distinct variable requirements. (Revised by Mario Carneiro, 19-Dec-2013.) |
Ref | Expression |
---|---|
eloprabga.1 |
Ref | Expression |
---|---|
eloprabga |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2867 | . 2 | |
2 | elex 2867 | . 2 | |
3 | elex 2867 | . 2 | |
4 | opexg 4587 | . . . . 5 | |
5 | opexg 4587 | . . . . 5 | |
6 | 4, 5 | sylan 457 | . . . 4 |
7 | 6 | 3impa 1146 | . . 3 |
8 | eqeq1 2359 | . . . . . . . . . . 11 | |
9 | eqcom 2355 | . . . . . . . . . . . 12 | |
10 | opth 4602 | . . . . . . . . . . . . . 14 | |
11 | 10 | anbi1i 676 | . . . . . . . . . . . . 13 |
12 | opth 4602 | . . . . . . . . . . . . 13 | |
13 | df-3an 936 | . . . . . . . . . . . . 13 | |
14 | 11, 12, 13 | 3bitr4i 268 | . . . . . . . . . . . 12 |
15 | 9, 14 | bitri 240 | . . . . . . . . . . 11 |
16 | 8, 15 | syl6bb 252 | . . . . . . . . . 10 |
17 | 16 | anbi1d 685 | . . . . . . . . 9 |
18 | eloprabga.1 | . . . . . . . . . 10 | |
19 | 18 | pm5.32i 618 | . . . . . . . . 9 |
20 | 17, 19 | syl6bb 252 | . . . . . . . 8 |
21 | 20 | 3exbidv 1629 | . . . . . . 7 |
22 | 21 | adantl 452 | . . . . . 6 |
23 | df-oprab 5528 | . . . . . . . . . 10 | |
24 | 23 | eleq2i 2417 | . . . . . . . . 9 |
25 | abid 2341 | . . . . . . . . 9 | |
26 | 24, 25 | bitr2i 241 | . . . . . . . 8 |
27 | eleq1 2413 | . . . . . . . 8 | |
28 | 26, 27 | syl5bb 248 | . . . . . . 7 |
29 | 28 | adantl 452 | . . . . . 6 |
30 | isset 2863 | . . . . . . . . . . . 12 | |
31 | isset 2863 | . . . . . . . . . . . 12 | |
32 | isset 2863 | . . . . . . . . . . . 12 | |
33 | 30, 31, 32 | 3anbi123i 1140 | . . . . . . . . . . 11 |
34 | eeeanv 1914 | . . . . . . . . . . 11 | |
35 | 33, 34 | bitr4i 243 | . . . . . . . . . 10 |
36 | 35 | biimpi 186 | . . . . . . . . 9 |
37 | 36 | biantrurd 494 | . . . . . . . 8 |
38 | 19.41vvv 1903 | . . . . . . . 8 | |
39 | 37, 38 | syl6rbbr 255 | . . . . . . 7 |
40 | 39 | adantr 451 | . . . . . 6 |
41 | 22, 29, 40 | 3bitr3d 274 | . . . . 5 |
42 | 41 | expcom 424 | . . . 4 |
43 | 42 | vtocleg 2925 | . . 3 |
44 | 7, 43 | mpcom 32 | . 2 |
45 | 1, 2, 3, 44 | syl3an 1224 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 w3a 934 wex 1541 wceq 1642 wcel 1710 cab 2339 cvv 2859 cop 4561 coprab 5527 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-oprab 5528 |
This theorem is referenced by: eloprabg 5579 ovigg 5596 |
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