Theorem funiunfvdm 5325

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 5324. (Contributed by Jim Kingdon, 10-Jan-2019.)

Assertion

Ref	Expression
funiunfvdm	⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A))

Distinct variable groups:   x,A   x,𝐹

Proof of Theorem funiunfvdm

Step	Hyp	Ref	Expression
1		fniunfv 5324	. 2 ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ ran 𝐹)
2		imadmrn 4603	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
3		fndm 4922	. . . . 5 ⊢ (𝐹 Fn A → dom 𝐹 = A)
4	3	imaeq2d 4593	. . . 4 ⊢ (𝐹 Fn A → (𝐹 “ dom 𝐹) = (𝐹 “ A))
5	2, 4	syl5eqr 2069	. . 3 ⊢ (𝐹 Fn A → ran 𝐹 = (𝐹 “ A))
6	5	unieqd 3564	. 2 ⊢ (𝐹 Fn A → ∪ ran 𝐹 = ∪ (𝐹 “ A))
7	1, 6	eqtrd 2055	1 ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A))

Syntax hints:   → wi 4   = wceq 1228  ∪ cuni 3553  ∪ ciun 3630  dom cdm 4270  ran crn 4271   “ cima 4273   Fn wfn 4822  ‘cfv 4827

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-rex 2289  df-v 2536  df-sbc 2741  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-op 3358  df-uni 3554  df-iun 3632  df-br 3738  df-opab 3792  df-mpt 3793  df-id 4003  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-res 4282  df-ima 4283  df-iota 4792  df-fun 4829  df-fn 4830  df-fv 4835

This theorem is referenced by:  funiunfvdmf  5326  eluniimadm  5327