![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7565 | . . . 4 ⊢ 1st :V–onto→V | |
2 | fofn 6460 | . . . 4 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 1st Fn V |
4 | ffn 6382 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴) | |
5 | dffn2 6384 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) | |
6 | 4, 5 | sylib 219 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V) |
7 | fnfco 6411 | . . 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 5980 | . . 3 ⊢ ran (1st ∘ 𝐹) = ran (1st ↾ ran 𝐹) | |
10 | frn 6388 | . . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶)) | |
11 | ssres2 5762 | . . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶))) | |
12 | rnss 5691 | . . . . 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 7569 | . . . . 5 ⊢ (1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 | |
15 | frn 6388 | . . . . 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 | syl6ss 3901 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ 𝐵) |
18 | 9, 17 | syl5eqss 3936 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ∘ 𝐹) ⊆ 𝐵) |
19 | df-f 6229 | . 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 3437 ⊆ wss 3859 × cxp 5441 ran crn 5444 ↾ cres 5445 ∘ ccom 5447 Fn wfn 6220 ⟶wf 6221 –onto→wfo 6223 1st c1st 7543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-fo 6231 df-fv 6233 df-1st 7545 |
This theorem is referenced by: ruclem11 15426 ruclem12 15427 caubl 23594 |
Copyright terms: Public domain | W3C validator |