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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-charfun | Unicode version |
Description: Properties of the characteristic function on the class of the class . (Contributed by BJ, 15-Aug-2024.) |
Ref | Expression |
---|---|
bj-charfun.1 |
Ref | Expression |
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bj-charfun |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-charfun.1 | . . 3 | |
2 | fmelpw1o 13423 | . . . 4 | |
3 | 2 | a1i 9 | . . 3 |
4 | 1, 3 | fmpt3d 5624 | . 2 |
5 | inss1 3327 | . . . . 5 | |
6 | 5 | a1i 9 | . . . 4 |
7 | difssd 3234 | . . . 4 | |
8 | 6, 7 | unssd 3283 | . . 3 |
9 | elun 3248 | . . . . 5 | |
10 | simpr 109 | . . . . . . . . . . 11 | |
11 | 10 | elin1d 3296 | . . . . . . . . . 10 |
12 | 1 | adantr 274 | . . . . . . . . . . 11 |
13 | 1oex 6372 | . . . . . . . . . . . . 13 | |
14 | 0ex 4092 | . . . . . . . . . . . . 13 | |
15 | 13, 14 | ifelpwun 4444 | . . . . . . . . . . . 12 |
16 | 15 | a1i 9 | . . . . . . . . . . 11 |
17 | 12, 16 | fvmpt2d 5555 | . . . . . . . . . 10 |
18 | 11, 17 | mpdan 418 | . . . . . . . . 9 |
19 | 10 | elin2d 3297 | . . . . . . . . . 10 |
20 | 19 | iftrued 3512 | . . . . . . . . 9 |
21 | 18, 20 | eqtrd 2190 | . . . . . . . 8 |
22 | 1lt2o 6390 | . . . . . . . 8 | |
23 | 21, 22 | eqeltrdi 2248 | . . . . . . 7 |
24 | 23 | ex 114 | . . . . . 6 |
25 | simpr 109 | . . . . . . . . . . 11 | |
26 | 25 | eldifad 3113 | . . . . . . . . . 10 |
27 | 1 | adantr 274 | . . . . . . . . . . 11 |
28 | 15 | a1i 9 | . . . . . . . . . . 11 |
29 | 27, 28 | fvmpt2d 5555 | . . . . . . . . . 10 |
30 | 26, 29 | mpdan 418 | . . . . . . . . 9 |
31 | 25 | eldifbd 3114 | . . . . . . . . . 10 |
32 | 31 | iffalsed 3515 | . . . . . . . . 9 |
33 | 30, 32 | eqtrd 2190 | . . . . . . . 8 |
34 | 0lt2o 6389 | . . . . . . . 8 | |
35 | 33, 34 | eqeltrdi 2248 | . . . . . . 7 |
36 | 35 | ex 114 | . . . . . 6 |
37 | 24, 36 | jaod 707 | . . . . 5 |
38 | 9, 37 | syl5bi 151 | . . . 4 |
39 | 38 | imp 123 | . . 3 |
40 | 4, 8, 39 | resflem 5632 | . 2 |
41 | 21 | ralrimiva 2530 | . . 3 |
42 | 33 | ralrimiva 2530 | . . 3 |
43 | 41, 42 | jca 304 | . 2 |
44 | 4, 40, 43 | jca31 307 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 698 wceq 1335 wcel 2128 wral 2435 cdif 3099 cun 3100 cin 3101 wss 3102 c0 3394 cif 3505 cpw 3543 cmpt 4026 cres 4589 wf 5167 cfv 5171 c1o 6357 c2o 6358 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-suc 4332 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-res 4599 df-ima 4600 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-1o 6364 df-2o 6365 |
This theorem is referenced by: bj-charfundcALT 13426 |
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