Theorem elunirn 5326

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 3553	. 2 ⊢ (A ∈ ∪ ran 𝐹 ↔ ∃y(A ∈ y ∧ y ∈ ran 𝐹))
2		funfn 4853	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
3		fvelrnb 5142	. . . . . . . 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 440	. . . . . 6 ⊢ (Fun 𝐹 → ((A ∈ y ∧ y ∈ ran 𝐹) ↔ (A ∈ y ∧ ∃x ∈ dom 𝐹(𝐹‘x) = y)))
6		r19.42v 2441	. . . . . 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 2079	. . . . . . 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 2390	. . . . 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 1678	. . 3 ⊢ (Fun 𝐹 → (∃y(A ∈ y ∧ y ∈ ran 𝐹) → ∃x ∈ dom 𝐹 A ∈ (𝐹‘x)))
13		fvelrn 5219	. . . . 5 ⊢ ((Fun 𝐹 ∧ x ∈ dom 𝐹) → (𝐹‘x) ∈ ran 𝐹)
14		funfvex 5113	. . . . . 6 ⊢ ((Fun 𝐹 ∧ x ∈ dom 𝐹) → (𝐹‘x) ∈ V)
15		eleq2 2079	. . . . . . . 8 ⊢ (y = (𝐹‘x) → (A ∈ y ↔ A ∈ (𝐹‘x)))
16		eleq1 2078	. . . . . . . 8 ⊢ (y = (𝐹‘x) → (y ∈ ran 𝐹 ↔ (𝐹‘x) ∈ ran 𝐹))
17	15, 16	anbi12d 445	. . . . . . 7 ⊢ (y = (𝐹‘x) → ((A ∈ y ∧ y ∈ ran 𝐹) ↔ (A ∈ (𝐹‘x) ∧ (𝐹‘x) ∈ ran 𝐹)))
18	17	spcegv 2614	. . . . . 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 406	. . . 4 ⊢ ((Fun 𝐹 ∧ x ∈ dom 𝐹) → (A ∈ (𝐹‘x) → ∃y(A ∈ y ∧ y ∈ ran 𝐹)))
21	20	rexlimdva 2407	. . 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 1226  ∃wex 1358   ∈ wcel 1370  ∃wrex 2281  Vcvv 2531  ∪ cuni 3550  dom cdm 4268  ran crn 4269  Fun wfun 4819   Fn wfn 4820  ‘cfv 4825

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833

This theorem is referenced by:  fnunirn  5327