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Mirrors > Home > NFE Home > Th. List > funiunfv | Unicode version |
Description: The indexed union of a
function's values is the union of its image under
the index class.
Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to , the theorem can be proved without this dependency. (Contributed by set.mm contributors, 26-Mar-2006.) |
Ref | Expression |
---|---|
funiunfv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 5328 | . . . . 5 | |
2 | eqid 2353 | . . . . 5 | |
3 | fvex 5339 | . . . . 5 | |
4 | 1, 2, 3 | fvopab4 5389 | . . . 4 |
5 | 4 | iuneq2i 3987 | . . 3 |
6 | fvex 5339 | . . . . 5 | |
7 | 6, 2 | fnopab2 5208 | . . . 4 |
8 | fniunfv 5466 | . . . 4 | |
9 | 7, 8 | ax-mp 8 | . . 3 |
10 | 5, 9 | eqtr3i 2375 | . 2 |
11 | rnopab2 4968 | . . . 4 | |
12 | 11 | unieqi 3901 | . . 3 |
13 | eqcom 2355 | . . . . . . . . 9 | |
14 | idd 21 | . . . . . . . . . 10 | |
15 | funbrfv 5356 | . . . . . . . . . . 11 | |
16 | 15 | adantr 451 | . . . . . . . . . 10 |
17 | n0i 3555 | . . . . . . . . . . . . 13 | |
18 | ndmfv 5349 | . . . . . . . . . . . . . . 15 | |
19 | eqeq1 2359 | . . . . . . . . . . . . . . 15 | |
20 | 18, 19 | syl5ib 210 | . . . . . . . . . . . . . 14 |
21 | 20 | con1d 116 | . . . . . . . . . . . . 13 |
22 | 17, 21 | mpan9 455 | . . . . . . . . . . . 12 |
23 | funbrfvb 5360 | . . . . . . . . . . . 12 | |
24 | 22, 23 | sylan2 460 | . . . . . . . . . . 11 |
25 | 24 | expr 598 | . . . . . . . . . 10 |
26 | 14, 16, 25 | pm5.21ndd 343 | . . . . . . . . 9 |
27 | 13, 26 | syl5bb 248 | . . . . . . . 8 |
28 | 27 | rexbidv 2635 | . . . . . . 7 |
29 | 28 | pm5.32da 622 | . . . . . 6 |
30 | 29 | exbidv 1626 | . . . . 5 |
31 | eluniab 3903 | . . . . 5 | |
32 | eluni 3894 | . . . . . 6 | |
33 | elima 4754 | . . . . . . . 8 | |
34 | 33 | anbi2i 675 | . . . . . . 7 |
35 | 34 | exbii 1582 | . . . . . 6 |
36 | 32, 35 | bitri 240 | . . . . 5 |
37 | 30, 31, 36 | 3bitr4g 279 | . . . 4 |
38 | 37 | eqrdv 2351 | . . 3 |
39 | 12, 38 | syl5eq 2397 | . 2 |
40 | 10, 39 | syl5eq 2397 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wa 358 wex 1541 wceq 1642 wcel 1710 cab 2339 wrex 2615 c0 3550 cuni 3891 ciun 3969 copab 4622 class class class wbr 4639 cima 4722 cdm 4772 crn 4773 wfun 4775 wfn 4776 cfv 4781 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-iun 3971 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-fv 4795 |
This theorem is referenced by: funiunfvf 5468 eluniima 5469 |
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