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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 5215 | . . 3 | |
2 | 1 | 3ad2ant1 1002 | . 2 |
3 | fnfun 5215 | . . 3 | |
4 | 3 | 3ad2ant2 1003 | . 2 |
5 | fndm 5217 | . . . . . . 7 | |
6 | fndm 5217 | . . . . . . 7 | |
7 | 5, 6 | ineqan12d 3274 | . . . . . 6 |
8 | 7 | eqeq1d 2146 | . . . . 5 |
9 | 8 | biimprd 157 | . . . 4 |
10 | 9 | adantrd 277 | . . 3 |
11 | 10 | 3impia 1178 | . 2 |
12 | simp3r 1010 | . . 3 | |
13 | 5 | eleq2d 2207 | . . . 4 |
14 | 13 | 3ad2ant1 1002 | . . 3 |
15 | 12, 14 | mpbird 166 | . 2 |
16 | funun 5162 | . . . . . . 7 | |
17 | ssun1 3234 | . . . . . . . . 9 | |
18 | dmss 4733 | . . . . . . . . 9 | |
19 | 17, 18 | ax-mp 5 | . . . . . . . 8 |
20 | 19 | sseli 3088 | . . . . . . 7 |
21 | 16, 20 | anim12i 336 | . . . . . 6 |
22 | 21 | anasss 396 | . . . . 5 |
23 | 22 | 3impa 1176 | . . . 4 |
24 | funfvdm 5477 | . . . 4 | |
25 | 23, 24 | syl 14 | . . 3 |
26 | imaundir 4947 | . . . . . 6 | |
27 | 26 | a1i 9 | . . . . 5 |
28 | 27 | unieqd 3742 | . . . 4 |
29 | disjel 3412 | . . . . . . . . 9 | |
30 | ndmima 4911 | . . . . . . . . 9 | |
31 | 29, 30 | syl 14 | . . . . . . . 8 |
32 | 31 | 3ad2ant3 1004 | . . . . . . 7 |
33 | 32 | uneq2d 3225 | . . . . . 6 |
34 | un0 3391 | . . . . . 6 | |
35 | 33, 34 | syl6eq 2186 | . . . . 5 |
36 | 35 | unieqd 3742 | . . . 4 |
37 | 28, 36 | eqtrd 2170 | . . 3 |
38 | funfvdm 5477 | . . . . . 6 | |
39 | 38 | eqcomd 2143 | . . . . 5 |
40 | 39 | adantrl 469 | . . . 4 |
41 | 40 | 3adant2 1000 | . . 3 |
42 | 25, 37, 41 | 3eqtrd 2174 | . 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 3064 cin 3065 wss 3066 c0 3358 csn 3522 cuni 3731 cdm 4534 cima 4537 wfun 5112 wfn 5113 cfv 5118 |
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 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
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 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 |
This theorem is referenced by: fvun2 5481 caseinl 6969 |
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