Theorem eldmrexrnb 7075

Description: For any element in the domain of a function, there is an element in the range of the function which is the value of the function at that element. Because of the definition df-fv 6531 of the value of a function, the theorem is only valid in general if the empty set is not contained in the range of the function (the implication "to the right" is always valid). Indeed, with the definition df-fv 6531 of the value of a function, (𝐹‘𝑌) = ∅ may mean that the value of 𝐹 at 𝑌 is the empty set or that 𝐹 is not defined at 𝑌. (Contributed by Alexander van der Vekens, 17-Dec-2017.)

Assertion

Ref	Expression
eldmrexrnb	⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))

Distinct variable groups:   𝑥,𝐹   𝑥,𝑌

Proof of Theorem eldmrexrnb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldmrexrn 7074	. . 3 ⊢ (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))
2	1	adantr 484	. 2 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))
3		eleq1 2852	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑌) → (𝑥 ∈ ran 𝐹 ↔ (𝐹‘𝑌) ∈ ran 𝐹))
4		elnelne2 3075	. . . . . . . . 9 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝐹‘𝑌) ≠ ∅)
5		n0 4307	. . . . . . . . . 10 ⊢ ((𝐹‘𝑌) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑌))
6		elfvdm 6903	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹)
7	6	exlimiv 1952	. . . . . . . . . 10 ⊢ (∃𝑦 𝑦 ∈ (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹)
8	5, 7	sylbi 219	. . . . . . . . 9 ⊢ ((𝐹‘𝑌) ≠ ∅ → 𝑌 ∈ dom 𝐹)
9	4, 8	syl 17	. . . . . . . 8 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹)
10	9	expcom 417	. . . . . . 7 ⊢ (∅ ∉ ran 𝐹 → ((𝐹‘𝑌) ∈ ran 𝐹 → 𝑌 ∈ dom 𝐹))
11	10	adantl 485	. . . . . 6 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → ((𝐹‘𝑌) ∈ ran 𝐹 → 𝑌 ∈ dom 𝐹))
12	11	com12 32	. . . . 5 ⊢ ((𝐹‘𝑌) ∈ ran 𝐹 → ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹))
13	3, 12	biimtrdi 255	. . . 4 ⊢ (𝑥 = (𝐹‘𝑌) → (𝑥 ∈ ran 𝐹 → ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹)))
14	13	com13 88	. . 3 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑥 ∈ ran 𝐹 → (𝑥 = (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹)))
15	14	rexlimdv 3163	. 2 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹))
16	2, 15	impbid 214	1 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144   ≠ wne 2959   ∉ wnel 3063  ∃wrex 3088  ∅c0 4287  dom cdm 5649  ran crn 5650  Fun wfun 6517  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-nel 3064  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-iota 6479  df-fun 6525  df-fn 6526  df-fv 6531

This theorem is referenced by: (None)