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Mirrors > Home > ILE Home > Th. List > funssfv | GIF version |
Description: The value of a member of the domain of a subclass of a function. (Contributed by NM, 15-Aug-1994.) |
Ref | Expression |
---|---|
funssfv | ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ A ∈ dom 𝐺) → (𝐹‘A) = (𝐺‘A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 5141 | . . . 4 ⊢ (A ∈ dom 𝐺 → ((𝐹 ↾ dom 𝐺)‘A) = (𝐹‘A)) | |
2 | 1 | eqcomd 2042 | . . 3 ⊢ (A ∈ dom 𝐺 → (𝐹‘A) = ((𝐹 ↾ dom 𝐺)‘A)) |
3 | funssres 4885 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺) | |
4 | 3 | fveq1d 5123 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺)‘A) = (𝐺‘A)) |
5 | 2, 4 | sylan9eqr 2091 | . 2 ⊢ (((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) ∧ A ∈ dom 𝐺) → (𝐹‘A) = (𝐺‘A)) |
6 | 5 | 3impa 1098 | 1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ A ∈ dom 𝐺) → (𝐹‘A) = (𝐺‘A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 ⊆ wss 2911 dom cdm 4288 ↾ cres 4290 Fun wfun 4839 ‘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-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-res 4300 df-iota 4810 df-fun 4847 df-fv 4853 |
This theorem is referenced by: tfrlem9 5876 tfrlemiubacc 5885 |
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