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Theorem funiunfvdm 5580
 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 5579. (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 5579 . 2 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
2 imadmrn 4817 . . . 4 (𝐹 “ dom 𝐹) = ran 𝐹
3 fndm 5147 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43imaeq2d 4807 . . . 4 (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
52, 4syl5eqr 2141 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹𝐴))
65unieqd 3686 . 2 (𝐹 Fn 𝐴 ran 𝐹 = (𝐹𝐴))
71, 6eqtrd 2127 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1296  ∪ cuni 3675  ∪ ciun 3752  dom cdm 4467  ran crn 4468   “ cima 4470   Fn wfn 5044  ‘cfv 5049 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060 This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-fv 5057 This theorem is referenced by:  funiunfvdmf  5581  eluniimadm  5582
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