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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 13784 | . . . . 5 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | 0cn 10468 | . . . . . 6 ⊢ 0 ∈ ℂ | |
3 | xpsng 6755 | . . . . . 6 ⊢ ((0 ∈ ℂ ∧ 𝑆 ∈ 𝐴) → ({0} × {𝑆}) = {〈0, 𝑆〉}) | |
4 | 2, 3 | mpan 686 | . . . . 5 ⊢ (𝑆 ∈ 𝐴 → ({0} × {𝑆}) = {〈0, 𝑆〉}) |
5 | 1, 4 | eqtr4d 2832 | . . . 4 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = ({0} × {𝑆})) |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 〈“𝑆”〉 = ({0} × {𝑆})) |
7 | 6 | coeq2d 5611 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 〈“𝑆”〉) = (𝐹 ∘ ({0} × {𝑆}))) |
8 | ffn 6374 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
9 | id 22 | . . . 4 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ 𝐴) | |
10 | fcoconst 6750 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝐴) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹‘𝑆)})) | |
11 | 8, 9, 10 | syl2anr 596 | . . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ ({0} × {𝑆})) = ({0} × {(𝐹‘𝑆)})) |
12 | fvex 6543 | . . . . 5 ⊢ (𝐹‘𝑆) ∈ V | |
13 | s1val 13784 | . . . . 5 ⊢ ((𝐹‘𝑆) ∈ V → 〈“(𝐹‘𝑆)”〉 = {〈0, (𝐹‘𝑆)〉}) | |
14 | 12, 13 | ax-mp 5 | . . . 4 ⊢ 〈“(𝐹‘𝑆)”〉 = {〈0, (𝐹‘𝑆)〉} |
15 | c0ex 10470 | . . . . 5 ⊢ 0 ∈ V | |
16 | 15, 12 | xpsn 6757 | . . . 4 ⊢ ({0} × {(𝐹‘𝑆)}) = {〈0, (𝐹‘𝑆)〉} |
17 | 14, 16 | eqtr4i 2820 | . . 3 ⊢ 〈“(𝐹‘𝑆)”〉 = ({0} × {(𝐹‘𝑆)}) |
18 | 11, 17 | syl6reqr 2848 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 〈“(𝐹‘𝑆)”〉 = (𝐹 ∘ ({0} × {𝑆}))) |
19 | 7, 18 | eqtr4d 2832 | 1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 〈“𝑆”〉) = 〈“(𝐹‘𝑆)”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 Vcvv 3432 {csn 4466 〈cop 4472 × cxp 5433 ∘ ccom 5439 Fn wfn 6212 ⟶wf 6213 ‘cfv 6217 ℂcc 10370 0cc0 10372 〈“cs1 13781 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-mulcl 10434 ax-i2m1 10440 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-reu 3110 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-s1 13782 |
This theorem is referenced by: cats1co 14042 s2co 14106 frmdgsum 17826 frmdup2 17829 efginvrel2 18568 vrgpinv 18610 frgpup2 18617 mrsubcv 32310 |
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