Theorem elunirn 5348

Description: Membership in the union of the range of a function. (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 3574	. 2 ⊢ (A ∈ ∪ ran 𝐹 ↔ ∃y(A ∈ y ∧ y ∈ ran 𝐹))
2		funfn 4874	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
3		fvelrnb 5164	. . . . . . . 8 ⊢ (𝐹 Fn dom 𝐹 → (y ∈ ran 𝐹 ↔ ∃x ∈ dom 𝐹(𝐹‘x) = y))
4	2, 3	sylbi 114	. . . . . . 7 ⊢ (Fun 𝐹 → (y ∈ ran 𝐹 ↔ ∃x ∈ dom 𝐹(𝐹‘x) = y))
5	4	anbi2d 437	. . . . . 6 ⊢ (Fun 𝐹 → ((A ∈ y ∧ y ∈ ran 𝐹) ↔ (A ∈ y ∧ ∃x ∈ dom 𝐹(𝐹‘x) = y)))
6		r19.42v 2461	. . . . . 6 ⊢ (∃x ∈ dom 𝐹(A ∈ y ∧ (𝐹‘x) = y) ↔ (A ∈ y ∧ ∃x ∈ dom 𝐹(𝐹‘x) = y))
7	5, 6	syl6bbr 187	. . . . 5 ⊢ (Fun 𝐹 → ((A ∈ y ∧ y ∈ ran 𝐹) ↔ ∃x ∈ dom 𝐹(A ∈ y ∧ (𝐹‘x) = y)))
8		eleq2 2098	. . . . . . 7 ⊢ ((𝐹‘x) = y → (A ∈ (𝐹‘x) ↔ A ∈ y))
9	8	biimparc 283	. . . . . 6 ⊢ ((A ∈ y ∧ (𝐹‘x) = y) → A ∈ (𝐹‘x))
10	9	reximi 2410	. . . . 5 ⊢ (∃x ∈ dom 𝐹(A ∈ y ∧ (𝐹‘x) = y) → ∃x ∈ dom 𝐹 A ∈ (𝐹‘x))
11	7, 10	syl6bi 152	. . . 4 ⊢ (Fun 𝐹 → ((A ∈ y ∧ y ∈ ran 𝐹) → ∃x ∈ dom 𝐹 A ∈ (𝐹‘x)))
12	11	exlimdv 1697	. . 3 ⊢ (Fun 𝐹 → (∃y(A ∈ y ∧ y ∈ ran 𝐹) → ∃x ∈ dom 𝐹 A ∈ (𝐹‘x)))
13		fvelrn 5241	. . . . 5 ⊢ ((Fun 𝐹 ∧ x ∈ dom 𝐹) → (𝐹‘x) ∈ ran 𝐹)
14		funfvex 5135	. . . . . 6 ⊢ ((Fun 𝐹 ∧ x ∈ dom 𝐹) → (𝐹‘x) ∈ V)
15		eleq2 2098	. . . . . . . 8 ⊢ (y = (𝐹‘x) → (A ∈ y ↔ A ∈ (𝐹‘x)))
16		eleq1 2097	. . . . . . . 8 ⊢ (y = (𝐹‘x) → (y ∈ ran 𝐹 ↔ (𝐹‘x) ∈ ran 𝐹))
17	15, 16	anbi12d 442	. . . . . . 7 ⊢ (y = (𝐹‘x) → ((A ∈ y ∧ y ∈ ran 𝐹) ↔ (A ∈ (𝐹‘x) ∧ (𝐹‘x) ∈ ran 𝐹)))
18	17	spcegv 2635	. . . . . 6 ⊢ ((𝐹‘x) ∈ V → ((A ∈ (𝐹‘x) ∧ (𝐹‘x) ∈ ran 𝐹) → ∃y(A ∈ y ∧ y ∈ ran 𝐹)))
19	14, 18	syl 14	. . . . 5 ⊢ ((Fun 𝐹 ∧ x ∈ dom 𝐹) → ((A ∈ (𝐹‘x) ∧ (𝐹‘x) ∈ ran 𝐹) → ∃y(A ∈ y ∧ y ∈ ran 𝐹)))
20	13, 19	mpan2d 404	. . . 4 ⊢ ((Fun 𝐹 ∧ x ∈ dom 𝐹) → (A ∈ (𝐹‘x) → ∃y(A ∈ y ∧ y ∈ ran 𝐹)))
21	20	rexlimdva 2427	. . 3 ⊢ (Fun 𝐹 → (∃x ∈ dom 𝐹 A ∈ (𝐹‘x) → ∃y(A ∈ y ∧ y ∈ ran 𝐹)))
22	12, 21	impbid 120	. 2 ⊢ (Fun 𝐹 → (∃y(A ∈ y ∧ y ∈ ran 𝐹) ↔ ∃x ∈ dom 𝐹 A ∈ (𝐹‘x)))
23	1, 22	syl5bb 181	1 ⊢ (Fun 𝐹 → (A ∈ ∪ ran 𝐹 ↔ ∃x ∈ dom 𝐹 A ∈ (𝐹‘x)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∃wrex 2301  Vcvv 2551  ∪ cuni 3571  dom cdm 4288  ran crn 4289  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by:  fnunirn  5349