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Mirrors > Home > MPE Home > Th. List > elrnrexdm | Structured version Visualization version GIF version |
Description: For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.) |
Ref | Expression |
---|---|
elrnrexdm | ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2824 | . . . . . 6 ⊢ (𝑌 ∈ ran 𝐹 → 𝑌 = 𝑌) | |
2 | 1 | ancli 551 | . . . . 5 ⊢ (𝑌 ∈ ran 𝐹 → (𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌)) |
3 | 2 | adantl 484 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ ran 𝐹) → (𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌)) |
4 | eqeq2 2835 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑌 = 𝑦 ↔ 𝑌 = 𝑌)) | |
5 | 4 | rspcev 3625 | . . . 4 ⊢ ((𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦) |
6 | 3, 5 | syl 17 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ ran 𝐹) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦) |
7 | 6 | ex 415 | . 2 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)) |
8 | funfn 6387 | . . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
9 | eqeq2 2835 | . . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑌 = 𝑦 ↔ 𝑌 = (𝐹‘𝑥))) | |
10 | 9 | rexrn 6855 | . . 3 ⊢ (𝐹 Fn dom 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥))) |
11 | 8, 10 | sylbi 219 | . 2 ⊢ (Fun 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥))) |
12 | 7, 11 | sylibd 241 | 1 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3141 dom cdm 5557 ran crn 5558 Fun wfun 6351 Fn wfn 6352 ‘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-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-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-mpt 5149 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: toprntopon 21535 wlkiswwlksupgr2 27657 loop1cycl 32386 bj-ccinftydisj 34497 gneispace 40491 |
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