Theorem elunirn 7002

Description: Membership in the union of the range of a function. See elunirnALT 7003 for a shorter proof which uses ax-pow 5234. (Contributed by NM, 24-Sep-2006.)

Assertion

Ref	Expression
elunirn	⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem elunirn

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4801	. 2 ⊢ (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹))
2		funfn 6365	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
3		fvelrnb 6714	. . . . . . . 8 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
4	2, 3	sylbi 220	. . . . . . 7 ⊢ (Fun 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
5	4	anbi2d 631	. . . . . 6 ⊢ (Fun 𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ 𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦)))
6		r19.42v 3268	. . . . . 6 ⊢ (∃𝑥 ∈ dom 𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝐴 ∈ 𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
7	5, 6	bitr4di 292	. . . . 5 ⊢ (Fun 𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦)))
8		eleq2 2840	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐴 ∈ (𝐹‘𝑥) ↔ 𝐴 ∈ 𝑦))
9	8	biimparc 483	. . . . . 6 ⊢ ((𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) → 𝐴 ∈ (𝐹‘𝑥))
10	9	reximi 3171	. . . . 5 ⊢ (∃𝑥 ∈ dom 𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))
11	7, 10	syl6bi 256	. . . 4 ⊢ (Fun 𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
12	11	exlimdv 1934	. . 3 ⊢ (Fun 𝐹 → (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
13		fvelrn 6835	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
14	13	a1d 25	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹‘𝑥) → (𝐹‘𝑥) ∈ ran 𝐹))
15	14	ancld 554	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹‘𝑥) → (𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹)))
16		fvex 6671	. . . . . 6 ⊢ (𝐹‘𝑥) ∈ V
17		eleq2 2840	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ (𝐹‘𝑥)))
18		eleq1 2839	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
19	17, 18	anbi12d 633	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹)))
20	16, 19	spcev 3525	. . . . 5 ⊢ ((𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹) → ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹))
21	15, 20	syl6 35	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹‘𝑥) → ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹)))
22	21	rexlimdva 3208	. . 3 ⊢ (Fun 𝐹 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥) → ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹)))
23	12, 22	impbid 215	. 2 ⊢ (Fun 𝐹 → (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
24	1, 23	syl5bb 286	1 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3071  ∪ cuni 4798  dom cdm 5524  ran crn 5525  Fun wfun 6329   Fn wfn 6330  ‘cfv 6335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-iota 6294  df-fun 6337  df-fn 6338  df-fv 6343

This theorem is referenced by:  fnunirn  7004  fin23lem30  9802  ustn0  22921  elrnust  22925  ustbas  22928  metuval  23251  elunirn2  30512  metidval  31361  pstmval  31366  elunirnmbfm  31739  fourierdlem70  43184  fourierdlem71  43185  fourierdlem80  43194