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Theorem funiunfv 5467
 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.)
Assertion
Ref Expression
funiunfv
Distinct variable groups:   ,   ,

Proof of Theorem funiunfv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5328 . . . . 5
2 eqid 2353 . . . . 5
3 fvex 5339 . . . . 5
41, 2, 3fvopab4 5389 . . . 4
54iuneq2i 3987 . . 3
6 fvex 5339 . . . . 5
76, 2fnopab2 5208 . . . 4
8 fniunfv 5466 . . . 4
97, 8ax-mp 8 . . 3
105, 9eqtr3i 2375 . 2
11 rnopab2 4968 . . . 4
1211unieqi 3901 . . 3
13 eqcom 2355 . . . . . . . . 9
14 idd 21 . . . . . . . . . 10
15 funbrfv 5356 . . . . . . . . . . 11
1615adantr 451 . . . . . . . . . 10
17 n0i 3555 . . . . . . . . . . . . 13
18 ndmfv 5349 . . . . . . . . . . . . . . 15
19 eqeq1 2359 . . . . . . . . . . . . . . 15
2018, 19syl5ib 210 . . . . . . . . . . . . . 14
2120con1d 116 . . . . . . . . . . . . 13
2217, 21mpan9 455 . . . . . . . . . . . 12
23 funbrfvb 5360 . . . . . . . . . . . 12
2422, 23sylan2 460 . . . . . . . . . . 11
2524expr 598 . . . . . . . . . 10
2614, 16, 25pm5.21ndd 343 . . . . . . . . 9
2713, 26syl5bb 248 . . . . . . . 8
2827rexbidv 2635 . . . . . . 7
2928pm5.32da 622 . . . . . 6
3029exbidv 1626 . . . . 5
31 eluniab 3903 . . . . 5
32 eluni 3894 . . . . . 6
33 elima 4754 . . . . . . . 8
3433anbi2i 675 . . . . . . 7
3534exbii 1582 . . . . . 6
3632, 35bitri 240 . . . . 5
3730, 31, 363bitr4g 279 . . . 4
3837eqrdv 2351 . . 3
3912, 38syl5eq 2397 . 2
4010, 39syl5eq 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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