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Mirrors > Home > ILE Home > Th. List > 2ndcof | GIF version |
Description: Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.) |
Ref | Expression |
---|---|
2ndcof | ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo2nd 6024 | . . . 4 ⊢ 2nd :V–onto→V | |
2 | fofn 5317 | . . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 2nd Fn V |
4 | ffn 5242 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴) | |
5 | dffn2 5244 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) | |
6 | 4, 5 | sylib 121 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V) |
7 | fnfco 5267 | . . 3 ⊢ ((2nd Fn V ∧ 𝐹:𝐴⟶V) → (2nd ∘ 𝐹) Fn 𝐴) | |
8 | 3, 6, 7 | sylancr 410 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹) Fn 𝐴) |
9 | rnco 5015 | . . 3 ⊢ ran (2nd ∘ 𝐹) = ran (2nd ↾ ran 𝐹) | |
10 | frn 5251 | . . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶)) | |
11 | ssres2 4816 | . . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶))) | |
12 | rnss 4739 | . . . . 5 ⊢ ((2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)) → ran (2nd ↾ ran 𝐹) ⊆ ran (2nd ↾ (𝐵 × 𝐶))) | |
13 | 10, 11, 12 | 3syl 17 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ ran (2nd ↾ (𝐵 × 𝐶))) |
14 | f2ndres 6026 | . . . . 5 ⊢ (2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶 | |
15 | frn 5251 | . . . . 5 ⊢ ((2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶 → ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶) | |
16 | 14, 15 | ax-mp 5 | . . . 4 ⊢ ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶 |
17 | 13, 16 | sstrdi 3079 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ 𝐶) |
18 | 9, 17 | eqsstrid 3113 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ∘ 𝐹) ⊆ 𝐶) |
19 | df-f 5097 | . 2 ⊢ ((2nd ∘ 𝐹):𝐴⟶𝐶 ↔ ((2nd ∘ 𝐹) Fn 𝐴 ∧ ran (2nd ∘ 𝐹) ⊆ 𝐶)) | |
20 | 8, 18, 19 | sylanbrc 413 | 1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Vcvv 2660 ⊆ wss 3041 × cxp 4507 ran crn 4510 ↾ cres 4511 ∘ ccom 4513 Fn wfn 5088 ⟶wf 5089 –onto→wfo 5091 2nd c2nd 6005 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fo 5099 df-fv 5101 df-2nd 6007 |
This theorem is referenced by: (None) |
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