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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 2178 | . . . . . 6 ⊢ (𝑌 ∈ ran 𝐹 → 𝑌 = 𝑌) | |
2 | 1 | ancli 323 | . . . . 5 ⊢ (𝑌 ∈ ran 𝐹 → (𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌)) |
3 | 2 | adantl 277 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑌 ∈ ran 𝐹) → (𝑌 ∈ ran 𝐹 ∧ 𝑌 = 𝑌)) |
4 | eqeq2 2187 | . . . . 5 ⊢ (𝑦 = 𝑌 → (𝑌 = 𝑦 ↔ 𝑌 = 𝑌)) | |
5 | 4 | rspcev 2841 | . . . 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 115 | . 2 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)) |
8 | funfn 5245 | . . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
9 | eqeq2 2187 | . . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑌 = 𝑦 ↔ 𝑌 = (𝐹‘𝑥))) | |
10 | 9 | rexrn 5652 | . . 3 ⊢ (𝐹 Fn dom 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥))) |
11 | 8, 10 | sylbi 121 | . 2 ⊢ (Fun 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥))) |
12 | 7, 11 | sylibd 149 | 1 ⊢ (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹‘𝑥))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∃wrex 2456 dom cdm 4625 ran crn 4626 Fun wfun 5209 Fn wfn 5210 ‘cfv 5215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-iota 5177 df-fun 5217 df-fn 5218 df-fv 5223 |
This theorem is referenced by: cc2lem 7262 ennnfonelemrnh 12409 ennnfonelemf1 12411 exmidsbthrlem 14630 |
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