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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 8019 | . . . 4 ⊢ 1st :V–onto→V | |
2 | fofn 6818 | . . . 4 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 1st Fn V |
4 | ffn 6727 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴) | |
5 | dffn2 6729 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) | |
6 | 4, 5 | sylib 217 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V) |
7 | fnfco 6767 | . . 3 ⊢ ((1st Fn V ∧ 𝐹:𝐴⟶V) → (1st ∘ 𝐹) Fn 𝐴) | |
8 | 3, 6, 7 | sylancr 585 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹) Fn 𝐴) |
9 | rnco 6261 | . . 3 ⊢ ran (1st ∘ 𝐹) = ran (1st ↾ ran 𝐹) | |
10 | frn 6734 | . . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶)) | |
11 | ssres2 6014 | . . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (1st ↾ ran 𝐹) ⊆ (1st ↾ (𝐵 × 𝐶))) | |
12 | rnss 5945 | . . . . 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 8023 | . . . . 5 ⊢ (1st ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐵 | |
15 | frn 6734 | . . . . 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 3994 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ↾ ran 𝐹) ⊆ 𝐵) |
18 | 9, 17 | eqsstrid 4030 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (1st ∘ 𝐹) ⊆ 𝐵) |
19 | df-f 6557 | . 2 ⊢ ((1st ∘ 𝐹):𝐴⟶𝐵 ↔ ((1st ∘ 𝐹) Fn 𝐴 ∧ ran (1st ∘ 𝐹) ⊆ 𝐵)) | |
20 | 8, 18, 19 | sylanbrc 581 | 1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (1st ∘ 𝐹):𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Vcvv 3473 ⊆ wss 3949 × cxp 5680 ran crn 5683 ↾ cres 5684 ∘ ccom 5686 Fn wfn 6548 ⟶wf 6549 –onto→wfo 6551 1st c1st 7997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-fun 6555 df-fn 6556 df-f 6557 df-fo 6559 df-1st 7999 |
This theorem is referenced by: ruclem11 16224 ruclem12 16225 caubl 25256 |
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