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Theorem funiunfvdm 5664
Description: The indexed union of a function's values is the union of its image under the index class. This theorem is a slight variation of fniunfv 5663. (Contributed by Jim Kingdon, 10-Jan-2019.)
Assertion
Ref Expression
funiunfvdm (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem funiunfvdm
StepHypRef Expression
1 fniunfv 5663 . 2 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
2 imadmrn 4891 . . . 4 (𝐹 “ dom 𝐹) = ran 𝐹
3 fndm 5222 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43imaeq2d 4881 . . . 4 (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
52, 4syl5eqr 2186 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹𝐴))
65unieqd 3747 . 2 (𝐹 Fn 𝐴 ran 𝐹 = (𝐹𝐴))
71, 6eqtrd 2172 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331   cuni 3736   ciun 3813  dom cdm 4539  ran crn 4540  cima 4542   Fn wfn 5118  cfv 5123
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-fv 5131
This theorem is referenced by:  funiunfvdmf  5665  eluniimadm  5666
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