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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 4899 | . . . . . . 7 ⊢ dom (𝐹 ↾ 𝐴) = (𝐴 ∩ dom 𝐹) | |
2 | 1 | elin2 3305 | . . . . . 6 ⊢ (𝐵 ∈ dom (𝐹 ↾ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹)) |
3 | funres 5223 | . . . . . . . . 9 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐴)) | |
4 | fvelrn 5610 | . . . . . . . . 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 | df-ima 4611 | . . . . . . . . . 10 ⊢ (𝐹 “ 𝐴) = ran (𝐹 ↾ 𝐴) | |
7 | 6 | eleq2i 2231 | . . . . . . . . 9 ⊢ ((𝐹‘𝐵) ∈ (𝐹 “ 𝐴) ↔ (𝐹‘𝐵) ∈ ran (𝐹 ↾ 𝐴)) |
8 | fvres 5504 | . . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐴 → ((𝐹 ↾ 𝐴)‘𝐵) = (𝐹‘𝐵)) | |
9 | 8 | eleq1d 2233 | . . . . . . . . 9 ⊢ (𝐵 ∈ 𝐴 → (((𝐹 ↾ 𝐴)‘𝐵) ∈ ran (𝐹 ↾ 𝐴) ↔ (𝐹‘𝐵) ∈ ran (𝐹 ↾ 𝐴))) |
10 | 7, 9 | bitr4id 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 2135 dom cdm 4598 ran crn 4599 ↾ cres 4600 “ cima 4601 Fun wfun 5176 ‘cfv 5182 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 |
This theorem is referenced by: funfvima2 5711 fiintim 6885 caseinl 7047 caseinr 7048 ctssdccl 7067 suplocexprlemdisj 7652 suplocexprlemub 7655 ennnfonelemex 12290 ctinfomlemom 12303 txcnp 12818 |
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