Theorem fniunfv 5902

Description: The indexed union of a function's values is the union of its range. Compare Definition 5.4 of [Monk1] p. 50. (Contributed by NM, 27-Sep-2004.)

Assertion

Ref	Expression
fniunfv	⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem fniunfv

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfvex 5656	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
2	1	funfni 5432	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V)
3	2	ralrimiva 2605	. . 3 ⊢ (𝐹 Fn 𝐴 → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ V)
4		dfiun2g 4002	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ V → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
5	3, 4	syl 14	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
6		fnrnfv 5692	. . 3 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
7	6	unieqd 3904	. 2 ⊢ (𝐹 Fn 𝐴 → ∪ ran 𝐹 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)})
8	5, 7	eqtr4d 2267	1 ⊢ (𝐹 Fn 𝐴 → ∪ 𝑥 ∈ 𝐴 (𝐹‘𝑥) = ∪ ran 𝐹)

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2202  {cab 2217  ∀wral 2510  ∃wrex 2511  Vcvv 2802  ∪ cuni 3893  ∪ ciun 3970  ran crn 4726   Fn wfn 5321  ‘cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by:  funiunfvdm  5903  ennnfonelemfun  13037  ennnfonelemf1  13038