Theorem eldmrexrnb 7126

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 6581 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 6581 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 7125	. . 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 2832	. . . . 5 ⊢ (𝑥 = (𝐹‘𝑌) → (𝑥 ∈ ran 𝐹 ↔ (𝐹‘𝑌) ∈ ran 𝐹))
4		elnelne2 3064	. . . . . . . . 9 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝐹‘𝑌) ≠ ∅)
5		n0 4376	. . . . . . . . . 10 ⊢ ((𝐹‘𝑌) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑌))
6		elfvdm 6957	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹)
7	6	exlimiv 1929	. . . . . . . . . 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 3159	. 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 1537  ∃wex 1777   ∈ wcel 2108   ≠ wne 2946   ∉ wnel 3052  ∃wrex 3076  ∅c0 4352  dom cdm 5700  ran crn 5701  Fun wfun 6567  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by: (None)