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Mirrors > Home > ILE Home > Th. List > fvres | GIF version |
Description: The value of a restricted function. (Contributed by NM, 2-Aug-1994.) |
Ref | Expression |
---|---|
fvres | ⊢ (A ∈ B → ((𝐹 ↾ B)‘A) = (𝐹‘A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2554 | . . . . 5 ⊢ x ∈ V | |
2 | 1 | brres 4561 | . . . 4 ⊢ (A(𝐹 ↾ B)x ↔ (A𝐹x ∧ A ∈ B)) |
3 | 2 | rbaib 829 | . . 3 ⊢ (A ∈ B → (A(𝐹 ↾ B)x ↔ A𝐹x)) |
4 | 3 | iotabidv 4831 | . 2 ⊢ (A ∈ B → (℩xA(𝐹 ↾ B)x) = (℩xA𝐹x)) |
5 | df-fv 4853 | . 2 ⊢ ((𝐹 ↾ B)‘A) = (℩xA(𝐹 ↾ B)x) | |
6 | df-fv 4853 | . 2 ⊢ (𝐹‘A) = (℩xA𝐹x) | |
7 | 4, 5, 6 | 3eqtr4g 2094 | 1 ⊢ (A ∈ B → ((𝐹 ↾ B)‘A) = (𝐹‘A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 class class class wbr 3755 ↾ cres 4290 ℩cio 4808 ‘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-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-xp 4294 df-res 4300 df-iota 4810 df-fv 4853 |
This theorem is referenced by: funssfv 5142 feqresmpt 5170 fvreseq 5214 respreima 5238 ffvresb 5271 fnressn 5292 fressnfv 5293 fvresi 5299 fvunsng 5300 fvsnun1 5303 fvsnun2 5304 fsnunfv 5306 funfvima 5333 isoresbr 5392 isores3 5398 isoini2 5401 ovres 5582 ofres 5667 offres 5704 fo1stresm 5730 fo2ndresm 5731 fo2ndf 5790 f1o2ndf1 5791 smores 5848 smores2 5850 tfrlem1 5864 rdgival 5909 rdgon 5913 frec0g 5922 frecsuclem1 5926 frecsuclem2 5928 frecrdg 5931 addpiord 6300 mulpiord 6301 fseq1p1m1 8726 iseqfeq2 8906 |
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