Theorem elunirn 7245

Description: Membership in the union of the range of a function. See elunirnALT 7246 for a shorter proof which uses ax-pow 5356. See elfvunirn 6916 for a more general version of the reverse direction. (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 4905	. 2 ⊢ (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹))
2		funfn 6571	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
3		fvelrnb 6945	. . . . . . . 8 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
4	2, 3	sylbi 216	. . . . . . 7 ⊢ (Fun 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
5	4	anbi2d 628	. . . . . 6 ⊢ (Fun 𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ 𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦)))
6		r19.42v 3184	. . . . . 6 ⊢ (∃𝑥 ∈ dom 𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝐴 ∈ 𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
7	5, 6	bitr4di 289	. . . . 5 ⊢ (Fun 𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦)))
8		eleq2 2816	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐴 ∈ (𝐹‘𝑥) ↔ 𝐴 ∈ 𝑦))
9	8	biimparc 479	. . . . . 6 ⊢ ((𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) → 𝐴 ∈ (𝐹‘𝑥))
10	9	reximi 3078	. . . . 5 ⊢ (∃𝑥 ∈ dom 𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))
11	7, 10	biimtrdi 252	. . . 4 ⊢ (Fun 𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
12	11	exlimdv 1928	. . 3 ⊢ (Fun 𝐹 → (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
13		fvelrn 7071	. . . . . . 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 550	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹‘𝑥) → (𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹)))
16		fvex 6897	. . . . . 6 ⊢ (𝐹‘𝑥) ∈ V
17		eleq2 2816	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ (𝐹‘𝑥)))
18		eleq1 2815	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
19	17, 18	anbi12d 630	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹)))
20	16, 19	spcev 3590	. . . . 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 3149	. . 3 ⊢ (Fun 𝐹 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥) → ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹)))
23	12, 22	impbid 211	. 2 ⊢ (Fun 𝐹 → (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
24	1, 23	bitrid 283	1 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  ∃wrex 3064  ∪ cuni 4902  dom cdm 5669  ran crn 5670  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is referenced by:  elunirn2OLD  7247  fnunirn  7248  fin23lem30  10336  ustn0  24075  ustbas  24082  elunirnmbfm  33779  fourierdlem70  45446  fourierdlem71  45447  fourierdlem80  45456