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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 5295 | . . 3 | |
2 | 1 | 3ad2ant1 1013 | . 2 |
3 | fnfun 5295 | . . 3 | |
4 | 3 | 3ad2ant2 1014 | . 2 |
5 | fndm 5297 | . . . . . . 7 | |
6 | fndm 5297 | . . . . . . 7 | |
7 | 5, 6 | ineqan12d 3330 | . . . . . 6 |
8 | 7 | eqeq1d 2179 | . . . . 5 |
9 | 8 | biimprd 157 | . . . 4 |
10 | 9 | adantrd 277 | . . 3 |
11 | 10 | 3impia 1195 | . 2 |
12 | simp3r 1021 | . . 3 | |
13 | 5 | eleq2d 2240 | . . . 4 |
14 | 13 | 3ad2ant1 1013 | . . 3 |
15 | 12, 14 | mpbird 166 | . 2 |
16 | funun 5242 | . . . . . . 7 | |
17 | ssun1 3290 | . . . . . . . . 9 | |
18 | dmss 4810 | . . . . . . . . 9 | |
19 | 17, 18 | ax-mp 5 | . . . . . . . 8 |
20 | 19 | sseli 3143 | . . . . . . 7 |
21 | 16, 20 | anim12i 336 | . . . . . 6 |
22 | 21 | anasss 397 | . . . . 5 |
23 | 22 | 3impa 1189 | . . . 4 |
24 | funfvdm 5559 | . . . 4 | |
25 | 23, 24 | syl 14 | . . 3 |
26 | imaundir 5024 | . . . . . 6 | |
27 | 26 | a1i 9 | . . . . 5 |
28 | 27 | unieqd 3807 | . . . 4 |
29 | disjel 3469 | . . . . . . . . 9 | |
30 | ndmima 4988 | . . . . . . . . 9 | |
31 | 29, 30 | syl 14 | . . . . . . . 8 |
32 | 31 | 3ad2ant3 1015 | . . . . . . 7 |
33 | 32 | uneq2d 3281 | . . . . . 6 |
34 | un0 3448 | . . . . . 6 | |
35 | 33, 34 | eqtrdi 2219 | . . . . 5 |
36 | 35 | unieqd 3807 | . . . 4 |
37 | 28, 36 | eqtrd 2203 | . . 3 |
38 | funfvdm 5559 | . . . . . 6 | |
39 | 38 | eqcomd 2176 | . . . . 5 |
40 | 39 | adantrl 475 | . . . 4 |
41 | 40 | 3adant2 1011 | . . 3 |
42 | 25, 37, 41 | 3eqtrd 2207 | . 2 |
43 | 2, 4, 11, 15, 42 | syl112anc 1237 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 cun 3119 cin 3120 wss 3121 c0 3414 csn 3583 cuni 3796 cdm 4611 cima 4614 wfun 5192 wfn 5193 cfv 5198 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-fv 5206 |
This theorem is referenced by: fvun2 5563 caseinl 7068 |
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