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Mirrors > Home > MPE Home > Th. List > eldmrexrnb | Structured version Visualization version GIF version |
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 5812 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 5812 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.) |
Ref | Expression |
---|---|
eldmrexrnb | ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldmrexrn 6273 | . . 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 2676 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑌) → (𝑥 ∈ ran 𝐹 ↔ (𝐹‘𝑌) ∈ ran 𝐹)) | |
4 | elnelne2 2894 | . . . . . . . . 9 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝐹‘𝑌) ≠ ∅) | |
5 | n0 3890 | . . . . . . . . . 10 ⊢ ((𝐹‘𝑌) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑌)) | |
6 | elfvdm 6130 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹) | |
7 | 6 | exlimiv 1845 | . . . . . . . . . 10 ⊢ (∃𝑦 𝑦 ∈ (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹) |
8 | 5, 7 | sylbi 206 | . . . . . . . . 9 ⊢ ((𝐹‘𝑌) ≠ ∅ → 𝑌 ∈ dom 𝐹) |
9 | 4, 8 | syl 17 | . . . . . . . 8 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹) |
10 | 9 | expcom 450 | . . . . . . 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 | syl6bi 242 | . . . 4 ⊢ (𝑥 = (𝐹‘𝑌) → (𝑥 ∈ ran 𝐹 → ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹))) |
14 | 13 | com13 86 | . . 3 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑥 ∈ ran 𝐹 → (𝑥 = (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹))) |
15 | 14 | rexlimdv 3012 | . 2 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹)) |
16 | 2, 15 | impbid 201 | 1 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∉ wnel 2781 ∃wrex 2897 ∅c0 3874 dom cdm 5038 ran crn 5039 Fun wfun 5798 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: (None) |
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