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Mirrors > Home > ILE Home > Th. List > elrnrexdm | 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 2038 | . . . . . 6 ⊢ (𝑌 ∈ ran 𝐹 → 𝑌 = 𝑌) | |
2 | 1 | ancli 306 | . . . . 5 ⊢ (𝑌 ∈ ran 𝐹 → (𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌)) |
3 | 2 | adantl 262 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ ran 𝐹) → (𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌)) |
4 | eqeq2 2046 | . . . . 5 ⊢ (y = 𝑌 → (𝑌 = y ↔ 𝑌 = 𝑌)) | |
5 | 4 | rspcev 2650 | . . . 4 ⊢ ((𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌) → ∃y ∈ ran 𝐹 𝑌 = y) |
6 | 3, 5 | syl 14 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ ran 𝐹) → ∃y ∈ ran 𝐹 𝑌 = y) |
7 | 6 | ex 108 | . 2 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃y ∈ ran 𝐹 𝑌 = y)) |
8 | funfn 4874 | . . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
9 | eqeq2 2046 | . . . 4 ⊢ (y = (𝐹‘x) → (𝑌 = y ↔ 𝑌 = (𝐹‘x))) | |
10 | 9 | rexrn 5247 | . . 3 ⊢ (𝐹 Fn dom 𝐹 → (∃y ∈ ran 𝐹 𝑌 = y ↔ ∃x ∈ dom 𝐹 𝑌 = (𝐹‘x))) |
11 | 8, 10 | sylbi 114 | . 2 ⊢ (Fun 𝐹 → (∃y ∈ ran 𝐹 𝑌 = y ↔ ∃x ∈ dom 𝐹 𝑌 = (𝐹‘x))) |
12 | 7, 11 | sylibd 138 | 1 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃x ∈ dom 𝐹 𝑌 = (𝐹‘x))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 ∃wrex 2301 dom cdm 4288 ran crn 4289 Fun wfun 4839 Fn wfn 4840 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 |
This theorem is referenced by: (None) |
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