![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > elrnmptg | GIF version |
Description: Membership in the range of a function. (Contributed by NM, 27-Aug-2007.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
rnmpt.1 | ⊢ 𝐹 = (x ∈ A ↦ B) |
Ref | Expression |
---|---|
elrnmptg | ⊢ (∀x ∈ A B ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃x ∈ A 𝐶 = B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnmpt.1 | . . . 4 ⊢ 𝐹 = (x ∈ A ↦ B) | |
2 | 1 | rnmpt 4525 | . . 3 ⊢ ran 𝐹 = {y ∣ ∃x ∈ A y = B} |
3 | 2 | eleq2i 2101 | . 2 ⊢ (𝐶 ∈ ran 𝐹 ↔ 𝐶 ∈ {y ∣ ∃x ∈ A y = B}) |
4 | r19.29 2444 | . . . . 5 ⊢ ((∀x ∈ A B ∈ 𝑉 ∧ ∃x ∈ A 𝐶 = B) → ∃x ∈ A (B ∈ 𝑉 ∧ 𝐶 = B)) | |
5 | eleq1 2097 | . . . . . . . 8 ⊢ (𝐶 = B → (𝐶 ∈ 𝑉 ↔ B ∈ 𝑉)) | |
6 | 5 | biimparc 283 | . . . . . . 7 ⊢ ((B ∈ 𝑉 ∧ 𝐶 = B) → 𝐶 ∈ 𝑉) |
7 | elex 2560 | . . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
8 | 6, 7 | syl 14 | . . . . . 6 ⊢ ((B ∈ 𝑉 ∧ 𝐶 = B) → 𝐶 ∈ V) |
9 | 8 | rexlimivw 2423 | . . . . 5 ⊢ (∃x ∈ A (B ∈ 𝑉 ∧ 𝐶 = B) → 𝐶 ∈ V) |
10 | 4, 9 | syl 14 | . . . 4 ⊢ ((∀x ∈ A B ∈ 𝑉 ∧ ∃x ∈ A 𝐶 = B) → 𝐶 ∈ V) |
11 | 10 | ex 108 | . . 3 ⊢ (∀x ∈ A B ∈ 𝑉 → (∃x ∈ A 𝐶 = B → 𝐶 ∈ V)) |
12 | eqeq1 2043 | . . . . 5 ⊢ (y = 𝐶 → (y = B ↔ 𝐶 = B)) | |
13 | 12 | rexbidv 2321 | . . . 4 ⊢ (y = 𝐶 → (∃x ∈ A y = B ↔ ∃x ∈ A 𝐶 = B)) |
14 | 13 | elab3g 2687 | . . 3 ⊢ ((∃x ∈ A 𝐶 = B → 𝐶 ∈ V) → (𝐶 ∈ {y ∣ ∃x ∈ A y = B} ↔ ∃x ∈ A 𝐶 = B)) |
15 | 11, 14 | syl 14 | . 2 ⊢ (∀x ∈ A B ∈ 𝑉 → (𝐶 ∈ {y ∣ ∃x ∈ A y = B} ↔ ∃x ∈ A 𝐶 = B)) |
16 | 3, 15 | syl5bb 181 | 1 ⊢ (∀x ∈ A B ∈ 𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃x ∈ A 𝐶 = B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 {cab 2023 ∀wral 2300 ∃wrex 2301 Vcvv 2551 ↦ cmpt 3809 ran crn 4289 |
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-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-br 3756 df-opab 3810 df-mpt 3811 df-cnv 4296 df-dm 4298 df-rn 4299 |
This theorem is referenced by: elrnmpti 4530 fliftel 5376 |
Copyright terms: Public domain | W3C validator |