| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > 2ndcof | Structured version Visualization version GIF version | ||
| Description: Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.) |
| Ref | Expression |
|---|---|
| 2ndcof | ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fo2nd 8035 | . . . 4 ⊢ 2nd :V–onto→V | |
| 2 | fofn 6822 | . . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
| 3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 2nd Fn V |
| 4 | ffn 6736 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴) | |
| 5 | dffn2 6738 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) | |
| 6 | 4, 5 | sylib 218 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V) |
| 7 | fnfco 6773 | . . 3 ⊢ ((2nd Fn V ∧ 𝐹:𝐴⟶V) → (2nd ∘ 𝐹) Fn 𝐴) | |
| 8 | 3, 6, 7 | sylancr 587 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹) Fn 𝐴) |
| 9 | rnco 6272 | . . 3 ⊢ ran (2nd ∘ 𝐹) = ran (2nd ↾ ran 𝐹) | |
| 10 | frn 6743 | . . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶)) | |
| 11 | ssres2 6022 | . . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶))) | |
| 12 | rnss 5950 | . . . . 5 ⊢ ((2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶)) → ran (2nd ↾ ran 𝐹) ⊆ ran (2nd ↾ (𝐵 × 𝐶))) | |
| 13 | 10, 11, 12 | 3syl 18 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ ran (2nd ↾ (𝐵 × 𝐶))) |
| 14 | f2ndres 8039 | . . . . 5 ⊢ (2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶 | |
| 15 | frn 6743 | . . . . 5 ⊢ ((2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶 → ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶) | |
| 16 | 14, 15 | ax-mp 5 | . . . 4 ⊢ ran (2nd ↾ (𝐵 × 𝐶)) ⊆ 𝐶 |
| 17 | 13, 16 | sstrdi 3996 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ 𝐶) |
| 18 | 9, 17 | eqsstrid 4022 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ∘ 𝐹) ⊆ 𝐶) |
| 19 | df-f 6565 | . 2 ⊢ ((2nd ∘ 𝐹):𝐴⟶𝐶 ↔ ((2nd ∘ 𝐹) Fn 𝐴 ∧ ran (2nd ∘ 𝐹) ⊆ 𝐶)) | |
| 20 | 8, 18, 19 | sylanbrc 583 | 1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 Vcvv 3480 ⊆ wss 3951 × cxp 5683 ran crn 5686 ↾ cres 5687 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 –onto→wfo 6559 2nd c2nd 8013 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-2nd 8015 |
| This theorem is referenced by: axdc4lem 10495 |
| Copyright terms: Public domain | W3C validator |