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Theorem funiunfvdm 5740
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 5739. (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 5739 . 2 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
2 imadmrn 4961 . . . 4 (𝐹 “ dom 𝐹) = ran 𝐹
3 fndm 5295 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43imaeq2d 4951 . . . 4 (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
52, 4eqtr3id 2217 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹𝐴))
65unieqd 3805 . 2 (𝐹 Fn 𝐴 ran 𝐹 = (𝐹𝐴))
71, 6eqtrd 2203 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348   cuni 3794   ciun 3871  dom cdm 4609  ran crn 4610  cima 4612   Fn wfn 5191  cfv 5196
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-fv 5204
This theorem is referenced by:  funiunfvdmf  5741  eluniimadm  5742
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