Theorem funiunfvdm 5785

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 5784. (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 5784	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹)
2		imadmrn 4998	. . . 4 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
3		fndm 5334	. . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
4	3	imaeq2d 4988	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴))
5	2, 4	eqtr3id 2236	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹 “ 𝐴))
6	5	unieqd 3835	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ ran 𝐹 = ∪ (𝐹 “ 𝐴))
7	1, 6	eqtrd 2222	1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ (𝐹 “ 𝐴))

Syntax hints:   → wi 4   = wceq 1364  ∪ cuni 3824  ∪ ciun 3901  dom cdm 4644  ran crn 4645   “ cima 4647   Fn wfn 5230  ‘cfv 5235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243

This theorem is referenced by:  funiunfvdmf  5786  eluniimadm  5787