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Mirrors > Home > ILE Home > Th. List > f2ndf | GIF version |
Description: The 2nd (second member of an ordered pair) function restricted to a function 𝐹 is a function of 𝐹 into the codomain of 𝐹. (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
Ref | Expression |
---|---|
f2ndf | ⊢ (𝐹:A⟶B → (2nd ↾ 𝐹):𝐹⟶B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f2ndres 5729 | . . 3 ⊢ (2nd ↾ (A × B)):(A × B)⟶B | |
2 | fssxp 5001 | . . 3 ⊢ (𝐹:A⟶B → 𝐹 ⊆ (A × B)) | |
3 | fssres 5009 | . . 3 ⊢ (((2nd ↾ (A × B)):(A × B)⟶B ∧ 𝐹 ⊆ (A × B)) → ((2nd ↾ (A × B)) ↾ 𝐹):𝐹⟶B) | |
4 | 1, 2, 3 | sylancr 393 | . 2 ⊢ (𝐹:A⟶B → ((2nd ↾ (A × B)) ↾ 𝐹):𝐹⟶B) |
5 | resabs1 4583 | . . . . 5 ⊢ (𝐹 ⊆ (A × B) → ((2nd ↾ (A × B)) ↾ 𝐹) = (2nd ↾ 𝐹)) | |
6 | 2, 5 | syl 14 | . . . 4 ⊢ (𝐹:A⟶B → ((2nd ↾ (A × B)) ↾ 𝐹) = (2nd ↾ 𝐹)) |
7 | 6 | eqcomd 2042 | . . 3 ⊢ (𝐹:A⟶B → (2nd ↾ 𝐹) = ((2nd ↾ (A × B)) ↾ 𝐹)) |
8 | 7 | feq1d 4977 | . 2 ⊢ (𝐹:A⟶B → ((2nd ↾ 𝐹):𝐹⟶B ↔ ((2nd ↾ (A × B)) ↾ 𝐹):𝐹⟶B)) |
9 | 4, 8 | mpbird 156 | 1 ⊢ (𝐹:A⟶B → (2nd ↾ 𝐹):𝐹⟶B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ⊆ wss 2911 × cxp 4286 ↾ cres 4290 ⟶wf 4841 2nd c2nd 5708 |
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-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 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-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 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-f 4849 df-fv 4853 df-2nd 5710 |
This theorem is referenced by: fo2ndf 5790 f1o2ndf1 5791 |
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