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Mirrors > Home > ILE Home > Th. List > fvun1 | Unicode version |
Description: The value of a union when the argument is in the first domain. (Contributed by Scott Fenton, 29-Jun-2013.) |
Ref | Expression |
---|---|
fvun1 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnfun 4996 | . . 3 | |
2 | 1 | 3ad2ant1 925 | . 2 |
3 | fnfun 4996 | . . 3 | |
4 | 3 | 3ad2ant2 926 | . 2 |
5 | fndm 4998 | . . . . . . 7 | |
6 | fndm 4998 | . . . . . . 7 | |
7 | 5, 6 | ineqan12d 3140 | . . . . . 6 |
8 | 7 | eqeq1d 2048 | . . . . 5 |
9 | 8 | biimprd 147 | . . . 4 |
10 | 9 | adantrd 264 | . . 3 |
11 | 10 | 3impia 1101 | . 2 |
12 | simp3r 933 | . . 3 | |
13 | 5 | eleq2d 2107 | . . . 4 |
14 | 13 | 3ad2ant1 925 | . . 3 |
15 | 12, 14 | mpbird 156 | . 2 |
16 | funun 4944 | . . . . . . 7 | |
17 | ssun1 3106 | . . . . . . . . 9 | |
18 | dmss 4534 | . . . . . . . . 9 | |
19 | 17, 18 | ax-mp 7 | . . . . . . . 8 |
20 | 19 | sseli 2941 | . . . . . . 7 |
21 | 16, 20 | anim12i 321 | . . . . . 6 |
22 | 21 | anasss 379 | . . . . 5 |
23 | 22 | 3impa 1099 | . . . 4 |
24 | funfvdm 5236 | . . . 4 | |
25 | 23, 24 | syl 14 | . . 3 |
26 | imaundir 4737 | . . . . . 6 | |
27 | 26 | a1i 9 | . . . . 5 |
28 | 27 | unieqd 3591 | . . . 4 |
29 | disjel 3274 | . . . . . . . . 9 | |
30 | ndmima 4702 | . . . . . . . . 9 | |
31 | 29, 30 | syl 14 | . . . . . . . 8 |
32 | 31 | 3ad2ant3 927 | . . . . . . 7 |
33 | 32 | uneq2d 3097 | . . . . . 6 |
34 | un0 3251 | . . . . . 6 | |
35 | 33, 34 | syl6eq 2088 | . . . . 5 |
36 | 35 | unieqd 3591 | . . . 4 |
37 | 28, 36 | eqtrd 2072 | . . 3 |
38 | funfvdm 5236 | . . . . . 6 | |
39 | 38 | eqcomd 2045 | . . . . 5 |
40 | 39 | adantrl 447 | . . . 4 |
41 | 40 | 3adant2 923 | . . 3 |
42 | 25, 37, 41 | 3eqtrd 2076 | . 2 |
43 | 2, 4, 11, 15, 42 | syl112anc 1139 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 97 wb 98 w3a 885 wceq 1243 wcel 1393 cun 2915 cin 2916 wss 2917 c0 3224 csn 3375 cuni 3580 cdm 4345 cima 4348 wfun 4896 wfn 4897 cfv 4902 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-fv 4910 |
This theorem is referenced by: fvun2 5240 |
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