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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 14120 | . . . . 5 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | 0cn 10790 | . . . . . 6 ⊢ 0 ∈ ℂ | |
3 | xpsng 6932 | . . . . . 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 2774 | . . . 4 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = ({0} × {𝑆})) |
6 | 5 | adantr 484 | . . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 〈“𝑆”〉 = ({0} × {𝑆})) |
7 | 6 | coeq2d 5716 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 〈“𝑆”〉) = (𝐹 ∘ ({0} × {𝑆}))) |
8 | fvex 6708 | . . . . 5 ⊢ (𝐹‘𝑆) ∈ V | |
9 | s1val 14120 | . . . . 5 ⊢ ((𝐹‘𝑆) ∈ V → 〈“(𝐹‘𝑆)”〉 = {〈0, (𝐹‘𝑆)〉}) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ 〈“(𝐹‘𝑆)”〉 = {〈0, (𝐹‘𝑆)〉} |
11 | c0ex 10792 | . . . . 5 ⊢ 0 ∈ V | |
12 | 11, 8 | xpsn 6934 | . . . 4 ⊢ ({0} × {(𝐹‘𝑆)}) = {〈0, (𝐹‘𝑆)〉} |
13 | 10, 12 | eqtr4i 2762 | . . 3 ⊢ 〈“(𝐹‘𝑆)”〉 = ({0} × {(𝐹‘𝑆)}) |
14 | ffn 6523 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
15 | id 22 | . . . 4 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ 𝐴) | |
16 | fcoconst 6927 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹‘𝑆)})) | |
17 | 14, 15, 16 | syl2anr 600 | . . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹‘𝑆)})) |
18 | 13, 17 | eqtr4id 2790 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 〈“(𝐹‘𝑆)”〉 = (𝐹 ∘ ({0} × {𝑆}))) |
19 | 7, 18 | eqtr4d 2774 | 1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 〈“𝑆”〉) = 〈“(𝐹‘𝑆)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 {csn 4527 〈cop 4533 × cxp 5534 ∘ ccom 5540 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 ℂcc 10692 0cc0 10694 〈“cs1 14117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-mulcl 10756 ax-i2m1 10762 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-s1 14118 |
This theorem is referenced by: cats1co 14386 s2co 14450 frmdgsum 18243 frmdup2 18246 efginvrel2 19071 vrgpinv 19113 frgpup2 19120 mrsubcv 33139 |
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