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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 5220 | . . 3 | |
2 | 1 | 3ad2ant1 1002 | . 2 |
3 | fnfun 5220 | . . 3 | |
4 | 3 | 3ad2ant2 1003 | . 2 |
5 | fndm 5222 | . . . . . . 7 | |
6 | fndm 5222 | . . . . . . 7 | |
7 | 5, 6 | ineqan12d 3279 | . . . . . 6 |
8 | 7 | eqeq1d 2148 | . . . . 5 |
9 | 8 | biimprd 157 | . . . 4 |
10 | 9 | adantrd 277 | . . 3 |
11 | 10 | 3impia 1178 | . 2 |
12 | simp3r 1010 | . . 3 | |
13 | 5 | eleq2d 2209 | . . . 4 |
14 | 13 | 3ad2ant1 1002 | . . 3 |
15 | 12, 14 | mpbird 166 | . 2 |
16 | funun 5167 | . . . . . . 7 | |
17 | ssun1 3239 | . . . . . . . . 9 | |
18 | dmss 4738 | . . . . . . . . 9 | |
19 | 17, 18 | ax-mp 5 | . . . . . . . 8 |
20 | 19 | sseli 3093 | . . . . . . 7 |
21 | 16, 20 | anim12i 336 | . . . . . 6 |
22 | 21 | anasss 396 | . . . . 5 |
23 | 22 | 3impa 1176 | . . . 4 |
24 | funfvdm 5484 | . . . 4 | |
25 | 23, 24 | syl 14 | . . 3 |
26 | imaundir 4952 | . . . . . 6 | |
27 | 26 | a1i 9 | . . . . 5 |
28 | 27 | unieqd 3747 | . . . 4 |
29 | disjel 3417 | . . . . . . . . 9 | |
30 | ndmima 4916 | . . . . . . . . 9 | |
31 | 29, 30 | syl 14 | . . . . . . . 8 |
32 | 31 | 3ad2ant3 1004 | . . . . . . 7 |
33 | 32 | uneq2d 3230 | . . . . . 6 |
34 | un0 3396 | . . . . . 6 | |
35 | 33, 34 | syl6eq 2188 | . . . . 5 |
36 | 35 | unieqd 3747 | . . . 4 |
37 | 28, 36 | eqtrd 2172 | . . 3 |
38 | funfvdm 5484 | . . . . . 6 | |
39 | 38 | eqcomd 2145 | . . . . 5 |
40 | 39 | adantrl 469 | . . . 4 |
41 | 40 | 3adant2 1000 | . . 3 |
42 | 25, 37, 41 | 3eqtrd 2176 | . 2 |
43 | 2, 4, 11, 15, 42 | syl112anc 1220 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 cun 3069 cin 3070 wss 3071 c0 3363 csn 3527 cuni 3736 cdm 4539 cima 4542 wfun 5117 wfn 5118 cfv 5123 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: fvun2 5488 caseinl 6976 |
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