| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 14636 | . . . . 5 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
| 2 | 0cn 11253 | . . . . . 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 2780 | . . . 4 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = ({0} × {𝑆})) |
| 6 | 5 | adantr 480 | . . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 〈“𝑆”〉 = ({0} × {𝑆})) |
| 7 | 6 | coeq2d 5873 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 〈“𝑆”〉) = (𝐹 ∘ ({0} × {𝑆}))) |
| 8 | fvex 6919 | . . . . 5 ⊢ (𝐹‘𝑆) ∈ V | |
| 9 | s1val 14636 | . . . . 5 ⊢ ((𝐹‘𝑆) ∈ V → 〈“(𝐹‘𝑆)”〉 = {〈0, (𝐹‘𝑆)〉}) | |
| 10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ 〈“(𝐹‘𝑆)”〉 = {〈0, (𝐹‘𝑆)〉} |
| 11 | c0ex 11255 | . . . . 5 ⊢ 0 ∈ V | |
| 12 | 11, 8 | xpsn 7161 | . . . 4 ⊢ ({0} × {(𝐹‘𝑆)}) = {〈0, (𝐹‘𝑆)〉} |
| 13 | 10, 12 | eqtr4i 2768 | . . 3 ⊢ 〈“(𝐹‘𝑆)”〉 = ({0} × {(𝐹‘𝑆)}) |
| 14 | ffn 6736 | . . . 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 2796 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 〈“(𝐹‘𝑆)”〉 = (𝐹 ∘ ({0} × {𝑆}))) |
| 19 | 7, 18 | eqtr4d 2780 | 1 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 〈“𝑆”〉) = 〈“(𝐹‘𝑆)”〉) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 〈cop 4632 × cxp 5683 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 ℂcc 11153 0cc0 11155 〈“cs1 14633 |
| 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-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-mulcl 11217 ax-i2m1 11223 |
| 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-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 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-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-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-s1 14634 |
| This theorem is referenced by: cats1co 14895 s2co 14959 frmdgsum 18875 frmdup2 18878 efginvrel2 19745 vrgpinv 19787 frgpup2 19794 ccatws1f1olast 32937 mrsubcv 35515 |
| Copyright terms: Public domain | W3C validator |