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Mirrors > Home > ILE Home > Th. List > funfvima | GIF version |
Description: A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.) |
Ref | Expression |
---|---|
funfvima | ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 4810 | . . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹) | |
2 | 1 | elin2 3234 | . . . . . 6 ⊢ (𝐵 ∈ dom (𝐹 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹)) |
3 | funres 5134 | . . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)) | |
4 | fvelrn 5519 | . . . . . . . . 9 ⊢ ((Fun (𝐹 ↾ 𝐴) ∧ 𝐵 ∈ dom (𝐹 ↾ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴)) | |
5 | 3, 4 | sylan 281 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom (𝐹 ↾ 𝐴)) → ((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴)) |
6 | fvres 5413 | . . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝐵) = (𝐹‘𝐵)) | |
7 | 6 | eleq1d 2186 | . . . . . . . . 9 ⊢ (𝐵 ∈ 𝐴 → (((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴) ↔ (𝐹‘𝐵) ∈ ran (𝐹 ↾ 𝐴))) |
8 | df-ima 4522 | . . . . . . . . . 10 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
9 | 8 | eleq2i 2184 | . . . . . . . . 9 ⊢ ((𝐹‘𝐵) ∈ (𝐹 “ 𝐴) ↔ (𝐹‘𝐵) ∈ ran (𝐹 ↾ 𝐴)) |
10 | 7, 9 | syl6rbbr 198 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → ((𝐹‘𝐵) ∈ (𝐹 “ 𝐴) ↔ ((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴))) |
11 | 5, 10 | syl5ibrcom 156 | . . . . . . 7 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom (𝐹 ↾ 𝐴)) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) |
12 | 11 | ex 114 | . . . . . 6 ⊢ (Fun 𝐹 → (𝐵 ∈ dom (𝐹 ↾ 𝐴) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))) |
13 | 2, 12 | syl5bir 152 | . . . . 5 ⊢ (Fun 𝐹 → ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))) |
14 | 13 | expd 256 | . . . 4 ⊢ (Fun 𝐹 → (𝐵 ∈ 𝐴 → (𝐵 ∈ dom 𝐹 → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))))) |
15 | 14 | com12 30 | . . 3 ⊢ (𝐵 ∈ 𝐴 → (Fun 𝐹 → (𝐵 ∈ dom 𝐹 → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))))) |
16 | 15 | impd 252 | . 2 ⊢ (𝐵 ∈ 𝐴 → ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))) |
17 | 16 | pm2.43b 52 | 1 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1465 dom cdm 4509 ran crn 4510 ↾ cres 4511 “ cima 4512 Fun wfun 5087 ‘cfv 5093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-fv 5101 |
This theorem is referenced by: funfvima2 5618 fiintim 6785 caseinl 6944 caseinr 6945 ctssdccl 6964 suplocexprlemdisj 7496 suplocexprlemub 7499 ennnfonelemex 11838 ctinfomlemom 11851 txcnp 12351 |
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