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Mirrors > Home > ILE Home > Th. List > fvimacnvi | GIF version |
Description: A member of a preimage is a function value argument. (Contributed by NM, 4-May-2007.) |
Ref | Expression |
---|---|
fvimacnvi | ⊢ ((Fun 𝐹 ∧ A ∈ (◡𝐹 “ B)) → (𝐹‘A) ∈ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 3499 | . . 3 ⊢ (A ∈ (◡𝐹 “ B) → {A} ⊆ (◡𝐹 “ B)) | |
2 | funimass2 4920 | . . 3 ⊢ ((Fun 𝐹 ∧ {A} ⊆ (◡𝐹 “ B)) → (𝐹 “ {A}) ⊆ B) | |
3 | 1, 2 | sylan2 270 | . 2 ⊢ ((Fun 𝐹 ∧ A ∈ (◡𝐹 “ B)) → (𝐹 “ {A}) ⊆ B) |
4 | cnvimass 4631 | . . . . 5 ⊢ (◡𝐹 “ B) ⊆ dom 𝐹 | |
5 | 4 | sseli 2935 | . . . 4 ⊢ (A ∈ (◡𝐹 “ B) → A ∈ dom 𝐹) |
6 | funfvex 5135 | . . . . 5 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → (𝐹‘A) ∈ V) | |
7 | snssg 3491 | . . . . 5 ⊢ ((𝐹‘A) ∈ V → ((𝐹‘A) ∈ B ↔ {(𝐹‘A)} ⊆ B)) | |
8 | 6, 7 | syl 14 | . . . 4 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → ((𝐹‘A) ∈ B ↔ {(𝐹‘A)} ⊆ B)) |
9 | 5, 8 | sylan2 270 | . . 3 ⊢ ((Fun 𝐹 ∧ A ∈ (◡𝐹 “ B)) → ((𝐹‘A) ∈ B ↔ {(𝐹‘A)} ⊆ B)) |
10 | funfn 4874 | . . . . . 6 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹) | |
11 | fnsnfv 5175 | . . . . . 6 ⊢ ((𝐹 Fn dom 𝐹 ∧ A ∈ dom 𝐹) → {(𝐹‘A)} = (𝐹 “ {A})) | |
12 | 10, 11 | sylanb 268 | . . . . 5 ⊢ ((Fun 𝐹 ∧ A ∈ dom 𝐹) → {(𝐹‘A)} = (𝐹 “ {A})) |
13 | 5, 12 | sylan2 270 | . . . 4 ⊢ ((Fun 𝐹 ∧ A ∈ (◡𝐹 “ B)) → {(𝐹‘A)} = (𝐹 “ {A})) |
14 | 13 | sseq1d 2966 | . . 3 ⊢ ((Fun 𝐹 ∧ A ∈ (◡𝐹 “ B)) → ({(𝐹‘A)} ⊆ B ↔ (𝐹 “ {A}) ⊆ B)) |
15 | 9, 14 | bitrd 177 | . 2 ⊢ ((Fun 𝐹 ∧ A ∈ (◡𝐹 “ B)) → ((𝐹‘A) ∈ B ↔ (𝐹 “ {A}) ⊆ B)) |
16 | 3, 15 | mpbird 156 | 1 ⊢ ((Fun 𝐹 ∧ A ∈ (◡𝐹 “ B)) → (𝐹‘A) ∈ B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 Vcvv 2551 ⊆ wss 2911 {csn 3367 ◡ccnv 4287 dom cdm 4288 “ cima 4291 Fun wfun 4839 Fn wfn 4840 ‘cfv 4845 |
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-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 |
This theorem is referenced by: fvimacnv 5225 elpreima 5229 |
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