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 (𝐹‘x) = ∪ (𝐹 “ A))

Distinct variable groups:   x,A   x,𝐹

Proof of Theorem funiunfvdm

Step	Hyp	Ref	Expression
1		fniunfv 5314	. 2 ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ ran 𝐹)
2		imadmrn 4593	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
3		fndm 4912	. . . . 5 ⊢ (𝐹 Fn A → dom 𝐹 = A)
4	3	imaeq2d 4583	. . . 4 ⊢ (𝐹 Fn A → (𝐹 “ dom 𝐹) = (𝐹 “ A))
5	2, 4	syl5eqr 2059	. . 3 ⊢ (𝐹 Fn A → ran 𝐹 = (𝐹 “ A))
6	5	unieqd 3554	. 2 ⊢ (𝐹 Fn A → ∪ ran 𝐹 = ∪ (𝐹 “ A))
7	1, 6	eqtrd 2045	1 ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A))

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