Theorem eldmrexrnb 7044

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 6506 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 6506 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 7043	. . 3 ⊢ (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))
2	1	adantr 480	. 2 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))
3		eleq1 2824	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑌) → (𝑥 ∈ ran 𝐹 ↔ (𝐹‘𝑌) ∈ ran 𝐹))
4		elnelne2 3048	. . . . . . . . 9 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝐹‘𝑌) ≠ ∅)
5		n0 4293	. . . . . . . . . 10 ⊢ ((𝐹‘𝑌) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑌))
6		elfvdm 6874	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹)
7	6	exlimiv 1932	. . . . . . . . . 10 ⊢ (∃𝑦 𝑦 ∈ (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹)
8	5, 7	sylbi 217	. . . . . . . . 9 ⊢ ((𝐹‘𝑌) ≠ ∅ → 𝑌 ∈ dom 𝐹)
9	4, 8	syl 17	. . . . . . . 8 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹)
10	9	expcom 413	. . . . . . 7 ⊢ (∅ ∉ ran 𝐹 → ((𝐹‘𝑌) ∈ ran 𝐹 → 𝑌 ∈ dom 𝐹))
11	10	adantl 481	. . . . . 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 253	. . . 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 3136	. 2 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹))
16	2, 15	impbid 212	1 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932   ∉ wnel 3036  ∃wrex 3061  ∅c0 4273  dom cdm 5631  ran crn 5632  Fun wfun 6492  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506

This theorem is referenced by: (None)