Theorem elunirn 7278

Description: Membership in the union of the range of a function. See elunirnALT 7279 for a shorter proof which uses ax-pow 5331. See elfvunirn 6938 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 4873	. 2 ⊢ (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹))
2		funfn 6590	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
3		fvelrnb 6968	. . . . . . . 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 642	. . . . . 6 ⊢ (Fun 𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ 𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦)))
6		r19.42v 3198	. . . . . 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 2853	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐴 ∈ (𝐹‘𝑥) ↔ 𝐴 ∈ 𝑦))
9	8	biimparc 485	. . . . . 6 ⊢ ((𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) → 𝐴 ∈ (𝐹‘𝑥))
10	9	reximi 3104	. . . . 5 ⊢ (∃𝑥 ∈ dom 𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))
11	7, 10	biimtrdi 256	. . . 4 ⊢ (Fun 𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
12	11	exlimdv 1968	. . 3 ⊢ (Fun 𝐹 → (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
13		fvelrn 7100	. . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
14	13	a1d 26	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹‘𝑥) → (𝐹‘𝑥) ∈ ran 𝐹))
15	14	ancld 560	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹‘𝑥) → (𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹)))
16		fvex 6921	. . . . . 6 ⊢ (𝐹‘𝑥) ∈ V
17		eleq2 2853	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ (𝐹‘𝑥)))
18		eleq1 2852	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
19	17, 18	anbi12d 644	. . . . . 6 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹)))
20	16, 19	spcev 3564	. . . . 5 ⊢ ((𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹) → ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹))
21	15, 20	syl6 36	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹‘𝑥) → ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹)))
22	21	rexlimdva 3167	. . 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	bitrid 286	1 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090  ∪ cuni 4870  dom cdm 5655  ran crn 5656  Fun wfun 6552   Fn wfn 6553  ‘cfv 6558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-fv 6566

This theorem is used by:  fnunirn  7280  fin23lem30  10447  ustn0  24570  ustbas  24576  elunirnmbfm  34955  fourierdlem70  47391  fourierdlem71  47392  fourierdlem80  47401