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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 7079 | . . . 4 ⊢ 1st :V–onto→V | |
2 | fofn 6030 | . . . 4 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 1st Fn V |
4 | ffn 5958 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴) | |
5 | dffn2 5960 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) | |
6 | 4, 5 | sylib 207 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V) |
7 | fnfco 5982 | . . 3 ⊢ ((1st Fn V ∧ 𝐹:𝐴⟶V) → (1st ∘ 𝐹) Fn 𝐴) | |
8 | 3, 6, 7 | sylancr 694 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹) Fn 𝐴) |
9 | rnco 5558 | . . 3 ⊢ ran (1st ∘ 𝐹) = ran (1st ↾ ran 𝐹) | |
10 | frn 5966 | . . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶)) | |
11 | ssres2 5345 | . . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶))) | |
12 | rnss 5275 | . . . . 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 7081 | . . . . 5 ⊢ (1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 | |
15 | frn 5966 | . . . . 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 3580 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ 𝐵) |
18 | 9, 17 | syl5eqss 3612 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ∘ 𝐹) ⊆ 𝐵) |
19 | df-f 5808 | . 2 ⊢ ((1st ∘ 𝐹):𝐴⟶𝐵 ↔ ((1st ∘ 𝐹) Fn 𝐴 ∧ ran (1st ∘ 𝐹) ⊆ 𝐵)) | |
20 | 8, 18, 19 | sylanbrc 695 | 1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹):𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Vcvv 3173 ⊆ wss 3540 × cxp 5036 ran crn 5039 ↾ cres 5040 ∘ ccom 5042 Fn wfn 5799 ⟶wf 5800 –onto→wfo 5802 1st c1st 7057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 |
This theorem is referenced by: ruclem11 14808 ruclem12 14809 caubl 22914 |
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