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Theorem funiunfvdm 5315
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 5314. (Contributed by Jim Kingdon, 10-Jan-2019.)
Assertion
Ref Expression
funiunfvdm (𝐹 Fn A x A (𝐹x) = (𝐹A))
Distinct variable groups:   x,A   x,𝐹

Proof of Theorem funiunfvdm
StepHypRef Expression
1 fniunfv 5314 . 2 (𝐹 Fn A x A (𝐹x) = ran 𝐹)
2 imadmrn 4593 . . . 4 (𝐹 “ dom 𝐹) = ran 𝐹
3 fndm 4912 . . . . 5 (𝐹 Fn A → dom 𝐹 = A)
43imaeq2d 4583 . . . 4 (𝐹 Fn A → (𝐹 “ dom 𝐹) = (𝐹A))
52, 4syl5eqr 2059 . . 3 (𝐹 Fn A → ran 𝐹 = (𝐹A))
65unieqd 3554 . 2 (𝐹 Fn A ran 𝐹 = (𝐹A))
71, 6eqtrd 2045 1 (𝐹 Fn A x A (𝐹x) = (𝐹A))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1223   cuni 3543   ciun 3620  dom cdm 4260  ran crn 4261  cima 4263   Fn wfn 4812  cfv 4817
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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907
This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-v 2528  df-sbc 2733  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-iun 3622  df-br 3728  df-opab 3782  df-mpt 3783  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-fv 4825
This theorem is referenced by:  funiunfvdmf  5316  eluniimadm  5317
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