Theorem elunirn 5767

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 3813	. 2 ⊢ (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹))
2		funfn 5247	. . . . . . . 8 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
3		fvelrnb 5564	. . . . . . . 8 ⊢ (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
4	2, 3	sylbi 121	. . . . . . 7 ⊢ (Fun 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
5	4	anbi2d 464	. . . . . 6 ⊢ (Fun 𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ 𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦)))
6		r19.42v 2634	. . . . . 6 ⊢ (∃𝑥 ∈ dom 𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) ↔ (𝐴 ∈ 𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹‘𝑥) = 𝑦))
7	5, 6	bitr4di 198	. . . . 5 ⊢ (Fun 𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦)))
8		eleq2 2241	. . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → (𝐴 ∈ (𝐹‘𝑥) ↔ 𝐴 ∈ 𝑦))
9	8	biimparc 299	. . . . . 6 ⊢ ((𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) → 𝐴 ∈ (𝐹‘𝑥))
10	9	reximi 2574	. . . . 5 ⊢ (∃𝑥 ∈ dom 𝐹(𝐴 ∈ 𝑦 ∧ (𝐹‘𝑥) = 𝑦) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥))
11	7, 10	syl6bi 163	. . . 4 ⊢ (Fun 𝐹 → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
12	11	exlimdv 1819	. . 3 ⊢ (Fun 𝐹 → (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
13		fvelrn 5648	. . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹)
14		funfvex 5533	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
15		eleq2 2241	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ (𝐹‘𝑥)))
16		eleq1 2240	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦 ∈ ran 𝐹 ↔ (𝐹‘𝑥) ∈ ran 𝐹))
17	15, 16	anbi12d 473	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ (𝐹‘𝑥) ∧ (𝐹‘𝑥) ∈ ran 𝐹)))
18	17	spcegv 2826	. . . . . 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 428	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹‘𝑥) → ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹)))
21	20	rexlimdva 2594	. . 3 ⊢ (Fun 𝐹 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥) → ∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹)))
22	12, 21	impbid 129	. 2 ⊢ (Fun 𝐹 → (∃𝑦(𝐴 ∈ 𝑦 ∧ 𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))
23	1, 22	bitrid 192	1 ⊢ (Fun 𝐹 → (𝐴 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1492   ∈ wcel 2148  ∃wrex 2456  Vcvv 2738  ∪ cuni 3810  dom cdm 4627  ran crn 4628  Fun wfun 5211   Fn wfn 5212  ‘cfv 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-fv 5225

This theorem is referenced by:  fnunirn  5768