Theorem funiunfvdm 5294

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 5293. (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 5293	. 2 ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ ran 𝐹)
2		imadmrn 4572	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
3		fndm 4891	. . . . 5 ⊢ (𝐹 Fn A → dom 𝐹 = A)
4	3	imaeq2d 4562	. . . 4 ⊢ (𝐹 Fn A → (𝐹 “ dom 𝐹) = (𝐹 “ A))
5	2, 4	syl5eqr 2068	. . 3 ⊢ (𝐹 Fn A → ran 𝐹 = (𝐹 “ A))
6	5	unieqd 3543	. 2 ⊢ (𝐹 Fn A → ∪ ran 𝐹 = ∪ (𝐹 “ A))
7	1, 6	eqtrd 2054	1 ⊢ (𝐹 Fn A → ∪ x ∈ A (𝐹‘x) = ∪ (𝐹 “ A))

Syntax hints:   → wi 4   = wceq 1373  ∪ cuni 3532  ∪ ciun 3609  dom cdm 4238  ran crn 4239   “ cima 4241   Fn wfn 4791  ‘cfv 4796

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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  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 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896

This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-eu 1884  df-mo 1885  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-rex 2288  df-v 2535  df-sbc 2740  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-op 3336  df-uni 3533  df-iun 3611  df-br 3717  df-opab 3771  df-mpt 3772  df-id 3983  df-xp 4244  df-rel 4245  df-cnv 4246  df-co 4247  df-dm 4248  df-rn 4249  df-res 4250  df-ima 4251  df-iota 4761  df-fun 4798  df-fn 4799  df-fv 4804

This theorem is referenced by:  funiunfvdmf  5295  eluniimadm  5296