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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 F Fn A, the theorem can be proved without this dependency. (Contributed by set.mm contributors, 26-Mar-2006.)
Assertion
Ref Expression
funiunfv (Fun Fx A (Fx) = (FA))
Distinct variable groups:   x,A   x,F

Proof of Theorem funiunfv
Dummy variables w y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5328 . . . . 5 (y = x → (Fy) = (Fx))
2 eqid 2353 . . . . 5 {y, z (y A z = (Fy))} = {y, z (y A z = (Fy))}
3 fvex 5339 . . . . 5 (Fx) V
41, 2, 3fvopab4 5389 . . . 4 (x A → ({y, z (y A z = (Fy))} ‘x) = (Fx))
54iuneq2i 3987 . . 3 x A ({y, z (y A z = (Fy))} ‘x) = x A (Fx)
6 fvex 5339 . . . . 5 (Fy) V
76, 2fnopab2 5208 . . . 4 {y, z (y A z = (Fy))} Fn A
8 fniunfv 5466 . . . 4 ({y, z (y A z = (Fy))} Fn Ax A ({y, z (y A z = (Fy))} ‘x) = ran {y, z (y A z = (Fy))})
97, 8ax-mp 8 . . 3 x A ({y, z (y A z = (Fy))} ‘x) = ran {y, z (y A z = (Fy))}
105, 9eqtr3i 2375 . 2 x A (Fx) = ran {y, z (y A z = (Fy))}
11 rnopab2 4968 . . . 4 ran {y, z (y A z = (Fy))} = {z y A z = (Fy)}
1211unieqi 3901 . . 3 ran {y, z (y A z = (Fy))} = {z y A z = (Fy)}
13 eqcom 2355 . . . . . . . . 9 (z = (Fy) ↔ (Fy) = z)
14 idd 21 . . . . . . . . . 10 ((Fun F w z) → ((Fy) = z → (Fy) = z))
15 funbrfv 5356 . . . . . . . . . . 11 (Fun F → (yFz → (Fy) = z))
1615adantr 451 . . . . . . . . . 10 ((Fun F w z) → (yFz → (Fy) = z))
17 n0i 3555 . . . . . . . . . . . . 13 (w z → ¬ z = )
18 ndmfv 5349 . . . . . . . . . . . . . . 15 y dom F → (Fy) = )
19 eqeq1 2359 . . . . . . . . . . . . . . 15 ((Fy) = z → ((Fy) = z = ))
2018, 19syl5ib 210 . . . . . . . . . . . . . 14 ((Fy) = z → (¬ y dom Fz = ))
2120con1d 116 . . . . . . . . . . . . 13 ((Fy) = z → (¬ z = y dom F))
2217, 21mpan9 455 . . . . . . . . . . . 12 ((w z (Fy) = z) → y dom F)
23 funbrfvb 5360 . . . . . . . . . . . 12 ((Fun F y dom F) → ((Fy) = zyFz))
2422, 23sylan2 460 . . . . . . . . . . 11 ((Fun F (w z (Fy) = z)) → ((Fy) = zyFz))
2524expr 598 . . . . . . . . . 10 ((Fun F w z) → ((Fy) = z → ((Fy) = zyFz)))
2614, 16, 25pm5.21ndd 343 . . . . . . . . 9 ((Fun F w z) → ((Fy) = zyFz))
2713, 26syl5bb 248 . . . . . . . 8 ((Fun F w z) → (z = (Fy) ↔ yFz))
2827rexbidv 2635 . . . . . . 7 ((Fun F w z) → (y A z = (Fy) ↔ y A yFz))
2928pm5.32da 622 . . . . . 6 (Fun F → ((w z y A z = (Fy)) ↔ (w z y A yFz)))
3029exbidv 1626 . . . . 5 (Fun F → (z(w z y A z = (Fy)) ↔ z(w z y A yFz)))
31 eluniab 3903 . . . . 5 (w {z y A z = (Fy)} ↔ z(w z y A z = (Fy)))
32 eluni 3894 . . . . . 6 (w (FA) ↔ z(w z z (FA)))
33 elima 4754 . . . . . . . 8 (z (FA) ↔ y A yFz)
3433anbi2i 675 . . . . . . 7 ((w z z (FA)) ↔ (w z y A yFz))
3534exbii 1582 . . . . . 6 (z(w z z (FA)) ↔ z(w z y A yFz))
3632, 35bitri 240 . . . . 5 (w (FA) ↔ z(w z y A yFz))
3730, 31, 363bitr4g 279 . . . 4 (Fun F → (w {z y A z = (Fy)} ↔ w (FA)))
3837eqrdv 2351 . . 3 (Fun F{z y A z = (Fy)} = (FA))
3912, 38syl5eq 2397 . 2 (Fun Fran {y, z (y A z = (Fy))} = (FA))
4010, 39syl5eq 2397 1 (Fun Fx A (Fx) = (FA))
 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  dom cdm 4772  ran crn 4773  Fun wfun 4775   Fn 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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