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Theorem funiunfvdm 5363
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 5362. (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 5362 . 2 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = ran 𝐹)
2 imadmrn 4639 . . . 4 (𝐹 “ dom 𝐹) = ran 𝐹
3 fndm 4959 . . . . 5 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
43imaeq2d 4629 . . . 4 (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹𝐴))
52, 4syl5eqr 2086 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹𝐴))
65unieqd 3587 . 2 (𝐹 Fn 𝐴 ran 𝐹 = (𝐹𝐴))
71, 6eqtrd 2072 1 (𝐹 Fn 𝐴 𝑥𝐴 (𝐹𝑥) = (𝐹𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243   cuni 3576   ciun 3653  dom cdm 4306  ran crn 4307  cima 4309   Fn wfn 4858  cfv 4863
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3871  ax-pow 3923  ax-pr 3940
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-sbc 2762  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3577  df-iun 3655  df-br 3761  df-opab 3815  df-mpt 3816  df-id 4026  df-xp 4312  df-rel 4313  df-cnv 4314  df-co 4315  df-dm 4316  df-rn 4317  df-res 4318  df-ima 4319  df-iota 4828  df-fun 4865  df-fn 4866  df-fv 4871
This theorem is referenced by:  funiunfvdmf  5364  eluniimadm  5365
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