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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 6365 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 6365 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 6859 | . . 3 ⊢ (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))) | |
2 | 1 | adantr 483 | . 2 ⊢ ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹‘𝑌))) |
3 | eleq1 2902 | . . . . 5 ⊢ (𝑥 = (𝐹‘𝑌) → (𝑥 ∈ ran 𝐹 ↔ (𝐹‘𝑌) ∈ ran 𝐹)) | |
4 | elnelne2 3136 | . . . . . . . . 9 ⊢ (((𝐹‘𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝐹‘𝑌) ≠ ∅) | |
5 | n0 4312 | . . . . . . . . . 10 ⊢ ((𝐹‘𝑌) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹‘𝑌)) | |
6 | elfvdm 6704 | . . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝐹‘𝑌) → 𝑌 ∈ dom 𝐹) | |
7 | 6 | exlimiv 1931 | . . . . . . . . . 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 416 | . . . . . . 7 ⊢ (∅ ∉ ran 𝐹 → ((𝐹‘𝑌) ∈ ran 𝐹 → 𝑌 ∈ dom 𝐹)) |
11 | 10 | adantl 484 | . . . . . 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 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 3285 | . 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))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ∉ wnel 3125 ∃wrex 3141 ∅c0 4293 dom cdm 5557 ran crn 5558 Fun wfun 6351 ‘cfv 6357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 |
This theorem is referenced by: (None) |
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