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Mirrors > Home > MPE Home > Th. List > s1co | Structured version Visualization version GIF version |
Description: Mapping of a singleton word. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 26-Feb-2016.) |
Ref | Expression |
---|---|
s1co | ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 〈“𝑆”〉) = 〈“(𝐹‘𝑆)”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | s1val 14633 | . . . . 5 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | 0cn 11251 | . . . . . 6 ⊢ 0 ∈ ℂ | |
3 | xpsng 7159 | . . . . . 6 ⊢ ((0 ∈ ℂ ∧ 𝑆 ∈ 𝐴) → ({0} × {𝑆}) = {〈0, 𝑆〉}) | |
4 | 2, 3 | mpan 690 | . . . . 5 ⊢ (𝑆 ∈ 𝐴 → ({0} × {𝑆}) = {〈0, 𝑆〉}) |
5 | 1, 4 | eqtr4d 2778 | . . . 4 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = ({0} × {𝑆})) |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 〈“𝑆”〉 = ({0} × {𝑆})) |
7 | 6 | coeq2d 5876 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 〈“𝑆”〉) = (𝐹 ∘ ({0} × {𝑆}))) |
8 | fvex 6920 | . . . . 5 ⊢ (𝐹‘𝑆) ∈ V | |
9 | s1val 14633 | . . . . 5 ⊢ ((𝐹‘𝑆) ∈ V → 〈“(𝐹‘𝑆)”〉 = {〈0, (𝐹‘𝑆)〉}) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ 〈“(𝐹‘𝑆)”〉 = {〈0, (𝐹‘𝑆)〉} |
11 | c0ex 11253 | . . . . 5 ⊢ 0 ∈ V | |
12 | 11, 8 | xpsn 7161 | . . . 4 ⊢ ({0} × {(𝐹‘𝑆)}) = {〈0, (𝐹‘𝑆)〉} |
13 | 10, 12 | eqtr4i 2766 | . . 3 ⊢ 〈“(𝐹‘𝑆)”〉 = ({0} × {(𝐹‘𝑆)}) |
14 | ffn 6737 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
15 | id 22 | . . . 4 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ 𝐴) | |
16 | fcoconst 7154 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹‘𝑆)})) | |
17 | 14, 15, 16 | syl2anr 597 | . . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹‘𝑆)})) |
18 | 13, 17 | eqtr4id 2794 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 〈“(𝐹‘𝑆)”〉 = (𝐹 ∘ ({0} × {𝑆}))) |
19 | 7, 18 | eqtr4d 2778 | 1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 〈“𝑆”〉) = 〈“(𝐹‘𝑆)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 {csn 4631 〈cop 4637 × cxp 5687 ∘ ccom 5693 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 ℂcc 11151 0cc0 11153 〈“cs1 14630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-mulcl 11215 ax-i2m1 11221 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-s1 14631 |
This theorem is referenced by: cats1co 14892 s2co 14956 frmdgsum 18888 frmdup2 18891 efginvrel2 19760 vrgpinv 19802 frgpup2 19809 ccatws1f1olast 32922 mrsubcv 35495 |
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