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Mirrors > Home > MPE Home > Th. List > 1stcof | Structured version Visualization version 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 7698 | . . . 4 ⊢ 1st :V–onto→V | |
2 | fofn 6585 | . . . 4 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 1st Fn V |
4 | ffn 6507 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴) | |
5 | dffn2 6509 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) | |
6 | 4, 5 | sylib 219 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V) |
7 | fnfco 6536 | . . 3 ⊢ ((1st Fn V ∧ 𝐹:𝐴⟶V) → (1st ∘ 𝐹) Fn 𝐴) | |
8 | 3, 6, 7 | sylancr 587 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹) Fn 𝐴) |
9 | rnco 6098 | . . 3 ⊢ ran (1st ∘ 𝐹) = ran (1st ↾ ran 𝐹) | |
10 | frn 6513 | . . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶)) | |
11 | ssres2 5874 | . . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶))) | |
12 | rnss 5802 | . . . . 5 ⊢ ((1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶)) → ran (1st ↾ ran 𝐹) ⊆ ran (1st ↾ (𝐵 × 𝐶))) | |
13 | 10, 11, 12 | 3syl 18 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ ran (1st ↾ (𝐵 × 𝐶))) |
14 | f1stres 7702 | . . . . 5 ⊢ (1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 | |
15 | frn 6513 | . . . . 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 3976 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ 𝐵) |
18 | 9, 17 | eqsstrid 4012 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ∘ 𝐹) ⊆ 𝐵) |
19 | df-f 6352 | . 2 ⊢ ((1st ∘ 𝐹):𝐴⟶𝐵 ↔ ((1st ∘ 𝐹) Fn 𝐴 ∧ ran (1st ∘ 𝐹) ⊆ 𝐵)) | |
20 | 8, 18, 19 | sylanbrc 583 | 1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹):𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Vcvv 3492 ⊆ wss 3933 × cxp 5546 ran crn 5549 ↾ cres 5550 ∘ ccom 5552 Fn wfn 6343 ⟶wf 6344 –onto→wfo 6346 1st c1st 7676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-1st 7678 |
This theorem is referenced by: ruclem11 15581 ruclem12 15582 caubl 23838 |
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