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Mirrors > Home > NFE Home > Th. List > funcnvuni | Unicode version |
Description: The union of a chain (with respect to inclusion) of single-rooted sets is single-rooted. (See funcnv 5156 for "single-rooted" definition.) (Contributed by set.mm contributors, 11-Aug-2004.) |
Ref | Expression |
---|---|
funcnvuni |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnveq 4886 | . . . . . . . 8 | |
2 | 1 | eqeq2d 2364 | . . . . . . 7 |
3 | 2 | cbvrexv 2836 | . . . . . 6 |
4 | cnveq 4886 | . . . . . . . . . . 11 | |
5 | 4 | funeqd 5129 | . . . . . . . . . 10 |
6 | sseq1 3292 | . . . . . . . . . . . 12 | |
7 | sseq2 3293 | . . . . . . . . . . . 12 | |
8 | 6, 7 | orbi12d 690 | . . . . . . . . . . 11 |
9 | 8 | ralbidv 2634 | . . . . . . . . . 10 |
10 | 5, 9 | anbi12d 691 | . . . . . . . . 9 |
11 | 10 | rspcv 2951 | . . . . . . . 8 |
12 | funeq 5127 | . . . . . . . . . 10 | |
13 | 12 | biimprcd 216 | . . . . . . . . 9 |
14 | sseq2 3293 | . . . . . . . . . . . . . . 15 | |
15 | sseq1 3292 | . . . . . . . . . . . . . . 15 | |
16 | 14, 15 | orbi12d 690 | . . . . . . . . . . . . . 14 |
17 | 16 | rspcv 2951 | . . . . . . . . . . . . 13 |
18 | cnvss 4885 | . . . . . . . . . . . . . . . 16 | |
19 | cnvss 4885 | . . . . . . . . . . . . . . . 16 | |
20 | 18, 19 | orim12i 502 | . . . . . . . . . . . . . . 15 |
21 | sseq12 3294 | . . . . . . . . . . . . . . . . 17 | |
22 | 21 | ancoms 439 | . . . . . . . . . . . . . . . 16 |
23 | sseq12 3294 | . . . . . . . . . . . . . . . 16 | |
24 | 22, 23 | orbi12d 690 | . . . . . . . . . . . . . . 15 |
25 | 20, 24 | syl5ibrcom 213 | . . . . . . . . . . . . . 14 |
26 | 25 | exp3a 425 | . . . . . . . . . . . . 13 |
27 | 17, 26 | syl6com 31 | . . . . . . . . . . . 12 |
28 | 27 | rexlimdv 2737 | . . . . . . . . . . 11 |
29 | 28 | com23 72 | . . . . . . . . . 10 |
30 | 29 | alrimdv 1633 | . . . . . . . . 9 |
31 | 13, 30 | anim12ii 553 | . . . . . . . 8 |
32 | 11, 31 | syl6com 31 | . . . . . . 7 |
33 | 32 | rexlimdv 2737 | . . . . . 6 |
34 | 3, 33 | syl5bi 208 | . . . . 5 |
35 | 34 | alrimiv 1631 | . . . 4 |
36 | df-ral 2619 | . . . . 5 | |
37 | vex 2862 | . . . . . . . 8 | |
38 | eqeq1 2359 | . . . . . . . . 9 | |
39 | 38 | rexbidv 2635 | . . . . . . . 8 |
40 | 37, 39 | elab 2985 | . . . . . . 7 |
41 | eqeq1 2359 | . . . . . . . . . 10 | |
42 | 41 | rexbidv 2635 | . . . . . . . . 9 |
43 | 42 | ralab 2997 | . . . . . . . 8 |
44 | 43 | anbi2i 675 | . . . . . . 7 |
45 | 40, 44 | imbi12i 316 | . . . . . 6 |
46 | 45 | albii 1566 | . . . . 5 |
47 | 36, 46 | bitr2i 241 | . . . 4 |
48 | 35, 47 | sylib 188 | . . 3 |
49 | fununi 5160 | . . 3 | |
50 | 48, 49 | syl 15 | . 2 |
51 | cnvuni 4895 | . . . 4 | |
52 | vex 2862 | . . . . . 6 | |
53 | 52 | cnvex 5102 | . . . . 5 |
54 | 53 | dfiun2 4001 | . . . 4 |
55 | 51, 54 | eqtri 2373 | . . 3 |
56 | 55 | funeqi 5128 | . 2 |
57 | 50, 56 | sylibr 203 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wo 357 wa 358 wal 1540 wceq 1642 wcel 1710 cab 2339 wral 2614 wrex 2615 wss 3257 cuni 3891 ciun 3969 ccnv 4771 wfun 4775 |
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-swap 4724 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-fun 4789 |
This theorem is referenced by: fun11uni 5162 |
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