Theorem funiunfvdm 5345

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 5344. (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 5344	. 2 ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ ran 𝐹)
2		imadmrn 4621	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
3		fndm 4941	. . . . 5 ⊢ (𝐹 Fn A → dom 𝐹 = A)
4	3	imaeq2d 4611	. . . 4 ⊢ (𝐹 Fn A → (𝐹 “ dom 𝐹) = (𝐹 “ A))
5	2, 4	syl5eqr 2083	. . 3 ⊢ (𝐹 Fn A → ran 𝐹 = (𝐹 “ A))
6	5	unieqd 3582	. 2 ⊢ (𝐹 Fn A → ∪ ran 𝐹 = ∪ (𝐹 “ A))
7	1, 6	eqtrd 2069	1 ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A))

Syntax hints:   → wi 4   = wceq 1242  ∪ cuni 3571  ∪ ciun 3648  dom cdm 4288  ran crn 4289   “ cima 4291   Fn wfn 4840  ‘cfv 4845

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by:  funiunfvdmf  5346  eluniimadm  5347