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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 5016 | . . 3 | |
2 | 1 | 3ad2ant1 959 | . 2 |
3 | fnfun 5016 | . . 3 | |
4 | 3 | 3ad2ant2 960 | . 2 |
5 | fndm 5018 | . . . . . . 7 | |
6 | fndm 5018 | . . . . . . 7 | |
7 | 5, 6 | ineqan12d 3169 | . . . . . 6 |
8 | 7 | eqeq1d 2089 | . . . . 5 |
9 | 8 | biimprd 156 | . . . 4 |
10 | 9 | adantrd 273 | . . 3 |
11 | 10 | 3impia 1135 | . 2 |
12 | simp3r 967 | . . 3 | |
13 | 5 | eleq2d 2148 | . . . 4 |
14 | 13 | 3ad2ant1 959 | . . 3 |
15 | 12, 14 | mpbird 165 | . 2 |
16 | funun 4964 | . . . . . . 7 | |
17 | ssun1 3135 | . . . . . . . . 9 | |
18 | dmss 4552 | . . . . . . . . 9 | |
19 | 17, 18 | ax-mp 7 | . . . . . . . 8 |
20 | 19 | sseli 2995 | . . . . . . 7 |
21 | 16, 20 | anim12i 331 | . . . . . 6 |
22 | 21 | anasss 391 | . . . . 5 |
23 | 22 | 3impa 1133 | . . . 4 |
24 | funfvdm 5257 | . . . 4 | |
25 | 23, 24 | syl 14 | . . 3 |
26 | imaundir 4757 | . . . . . 6 | |
27 | 26 | a1i 9 | . . . . 5 |
28 | 27 | unieqd 3612 | . . . 4 |
29 | disjel 3298 | . . . . . . . . 9 | |
30 | ndmima 4722 | . . . . . . . . 9 | |
31 | 29, 30 | syl 14 | . . . . . . . 8 |
32 | 31 | 3ad2ant3 961 | . . . . . . 7 |
33 | 32 | uneq2d 3126 | . . . . . 6 |
34 | un0 3278 | . . . . . 6 | |
35 | 33, 34 | syl6eq 2129 | . . . . 5 |
36 | 35 | unieqd 3612 | . . . 4 |
37 | 28, 36 | eqtrd 2113 | . . 3 |
38 | funfvdm 5257 | . . . . . 6 | |
39 | 38 | eqcomd 2086 | . . . . 5 |
40 | 39 | adantrl 461 | . . . 4 |
41 | 40 | 3adant2 957 | . . 3 |
42 | 25, 37, 41 | 3eqtrd 2117 | . 2 |
43 | 2, 4, 11, 15, 42 | syl112anc 1173 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 cun 2971 cin 2972 wss 2973 c0 3251 csn 3398 cuni 3601 cdm 4363 cima 4366 wfun 4916 wfn 4917 cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-fv 4930 |
This theorem is referenced by: fvun2 5261 |
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