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Mirrors > Home > ILE Home > Th. List > 1stcof | GIF version |
Description: Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.) |
Ref | Expression |
---|---|
1stcof | ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹):𝐴⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo1st 6105 | . . . 4 ⊢ 1st :V–onto→V | |
2 | fofn 5394 | . . . 4 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 1st Fn V |
4 | ffn 5319 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴) | |
5 | dffn2 5321 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) | |
6 | 4, 5 | sylib 121 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V) |
7 | fnfco 5344 | . . 3 ⊢ ((1st Fn V ∧ 𝐹:𝐴⟶V) → (1st ∘ 𝐹) Fn 𝐴) | |
8 | 3, 6, 7 | sylancr 411 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹) Fn 𝐴) |
9 | rnco 5092 | . . 3 ⊢ ran (1st ∘ 𝐹) = ran (1st ↾ ran 𝐹) | |
10 | frn 5328 | . . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶)) | |
11 | ssres2 4893 | . . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶))) | |
12 | rnss 4816 | . . . . 5 ⊢ ((1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶)) → ran (1st ↾ ran 𝐹) ⊆ ran (1st ↾ (𝐵 × 𝐶))) | |
13 | 10, 11, 12 | 3syl 17 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ ran (1st ↾ (𝐵 × 𝐶))) |
14 | f1stres 6107 | . . . . 5 ⊢ (1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 | |
15 | frn 5328 | . . . . 5 ⊢ ((1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 → ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵) | |
16 | 14, 15 | ax-mp 5 | . . . 4 ⊢ ran (1st ↾ (𝐵 × 𝐶)) ⊆ 𝐵 |
17 | 13, 16 | sstrdi 3140 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ 𝐵) |
18 | 9, 17 | eqsstrid 3174 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ∘ 𝐹) ⊆ 𝐵) |
19 | df-f 5174 | . 2 ⊢ ((1st ∘ 𝐹):𝐴⟶𝐵 ↔ ((1st ∘ 𝐹) Fn 𝐴 ∧ ran (1st ∘ 𝐹) ⊆ 𝐵)) | |
20 | 8, 18, 19 | sylanbrc 414 | 1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹):𝐴⟶𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 Vcvv 2712 ⊆ wss 3102 × cxp 4584 ran crn 4587 ↾ cres 4588 ∘ ccom 4590 Fn wfn 5165 ⟶wf 5166 –onto→wfo 5168 1st c1st 6086 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-fo 5176 df-fv 5178 df-1st 6088 |
This theorem is referenced by: (None) |
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