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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 7983 | . . . 4 ⊢ 2nd :V–onto→V | |
2 | fofn 6797 | . . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ 2nd Fn V |
4 | ffn 6707 | . . . 4 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹 Fn 𝐴) | |
5 | dffn2 6709 | . . . 4 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶V) | |
6 | 4, 5 | sylib 217 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → 𝐹:𝐴⟶V) |
7 | fnfco 6746 | . . 3 ⊢ ((2nd Fn V ∧ 𝐹:𝐴⟶V) → (2nd ∘ 𝐹) Fn 𝐴) | |
8 | 3, 6, 7 | sylancr 588 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹) Fn 𝐴) |
9 | rnco 6243 | . . 3 ⊢ ran (2nd ∘ 𝐹) = ran (2nd ↾ ran 𝐹) | |
10 | frn 6714 | . . . . 5 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran 𝐹 ⊆ (𝐵 × 𝐶)) | |
11 | ssres2 6004 | . . . . 5 ⊢ (ran 𝐹 ⊆ (𝐵 × 𝐶) → (2nd ↾ ran 𝐹) ⊆ (2nd ↾ (𝐵 × 𝐶))) | |
12 | rnss 5933 | . . . . 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 7987 | . . . . 5 ⊢ (2nd ↾ (𝐵 × 𝐶)):(𝐵 × 𝐶)⟶𝐶 | |
15 | frn 6714 | . . . . 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 3992 | . . 3 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ↾ ran 𝐹) ⊆ 𝐶) |
18 | 9, 17 | eqsstrid 4028 | . 2 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → ran (2nd ∘ 𝐹) ⊆ 𝐶) |
19 | df-f 6539 | . 2 ⊢ ((2nd ∘ 𝐹):𝐴⟶𝐶 ↔ ((2nd ∘ 𝐹) Fn 𝐴 ∧ ran (2nd ∘ 𝐹) ⊆ 𝐶)) | |
20 | 8, 18, 19 | sylanbrc 584 | 1 ⊢ (𝐹:𝐴⟶(𝐵 × 𝐶) → (2nd ∘ 𝐹):𝐴⟶𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Vcvv 3475 ⊆ wss 3946 × cxp 5670 ran crn 5673 ↾ cres 5674 ∘ ccom 5676 Fn wfn 6530 ⟶wf 6531 –onto→wfo 6533 2nd c2nd 7961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-fun 6537 df-fn 6538 df-f 6539 df-fo 6541 df-2nd 7963 |
This theorem is referenced by: axdc4lem 10437 |
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