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Theorem funiunfvdm 5480
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 5479. (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 5479 . 2 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
2 imadmrn 4737 . . . 4 (𝐹 “ dom 𝐹) = ran 𝐹
3 fndm 5064 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43imaeq2d 4727 . . . 4 (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
52, 4syl5eqr 2129 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹𝐴))
65unieqd 3638 . 2 (𝐹 Fn 𝐴 ran 𝐹 = (𝐹𝐴))
71, 6eqtrd 2115 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285   cuni 3627   ciun 3704  dom cdm 4399  ran crn 4400  cima 4402   Fn wfn 4962  cfv 4967
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-fv 4975
This theorem is referenced by:  funiunfvdmf  5481  eluniimadm  5482
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