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Mirrors > Home > ILE Home > Th. List > fvelrn | GIF version |
Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.) |
Ref | Expression |
---|---|
fvelrn | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2145 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹)) | |
2 | 1 | anbi2d 452 | . . . 4 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))) |
3 | fveq2 5251 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
4 | 3 | eleq1d 2151 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ran 𝐹 ↔ (𝐹‘𝐴) ∈ ran 𝐹)) |
5 | 2, 4 | imbi12d 232 | . . 3 ⊢ (𝑥 = 𝐴 → (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹))) |
6 | funfvop 5354 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) | |
7 | vex 2615 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | opeq1 3596 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → 〈𝑦, (𝐹‘𝑥)〉 = 〈𝑥, (𝐹‘𝑥)〉) | |
9 | 8 | eleq1d 2151 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹 ↔ 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹)) |
10 | 7, 9 | spcev 2703 | . . . . 5 ⊢ (〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹 → ∃𝑦〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹) |
11 | 6, 10 | syl 14 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ∃𝑦〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹) |
12 | funfvex 5265 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V) | |
13 | elrn2g 4582 | . . . . 5 ⊢ ((𝐹‘𝑥) ∈ V → ((𝐹‘𝑥) ∈ ran 𝐹 ↔ ∃𝑦〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹)) | |
14 | 12, 13 | syl 14 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ((𝐹‘𝑥) ∈ ran 𝐹 ↔ ∃𝑦〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹)) |
15 | 11, 14 | mpbird 165 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) |
16 | 5, 15 | vtoclg 2669 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)) |
17 | 16 | anabsi7 546 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∃wex 1422 ∈ wcel 1434 Vcvv 2612 〈cop 3425 dom cdm 4399 ran crn 4400 Fun wfun 4961 ‘cfv 4967 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-sbc 2827 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4083 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-iota 4932 df-fun 4969 df-fn 4970 df-fv 4975 |
This theorem is referenced by: fnfvelrn 5374 eldmrexrn 5383 funfvima 5464 elunirn 5483 frecuzrdgdomlem 9711 frecuzrdgsuctlem 9717 |
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