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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 14439 | . . . . 5 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = {〈0, 𝑆〉}) | |
2 | 0cn 11105 | . . . . . 6 ⊢ 0 ∈ ℂ | |
3 | xpsng 7081 | . . . . . 6 ⊢ ((0 ∈ ℂ ∧ 𝑆 ∈ 𝐴) → ({0} × {𝑆}) = {〈0, 𝑆〉}) | |
4 | 2, 3 | mpan 688 | . . . . 5 ⊢ (𝑆 ∈ 𝐴 → ({0} × {𝑆}) = {〈0, 𝑆〉}) |
5 | 1, 4 | eqtr4d 2780 | . . . 4 ⊢ (𝑆 ∈ 𝐴 → 〈“𝑆”〉 = ({0} × {𝑆})) |
6 | 5 | adantr 481 | . . 3 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → 〈“𝑆”〉 = ({0} × {𝑆})) |
7 | 6 | coeq2d 5816 | . 2 ⊢ ((𝑆 ∈ 𝐴 ∧ 𝐹:𝐴⟶𝐵) → (𝐹 ∘ 〈“𝑆”〉) = (𝐹 ∘ ({0} × {𝑆}))) |
8 | fvex 6852 | . . . . 5 ⊢ (𝐹‘𝑆) ∈ V | |
9 | s1val 14439 | . . . . 5 ⊢ ((𝐹‘𝑆) ∈ V → 〈“(𝐹‘𝑆)”〉 = {〈0, (𝐹‘𝑆)〉}) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ 〈“(𝐹‘𝑆)”〉 = {〈0, (𝐹‘𝑆)〉} |
11 | c0ex 11107 | . . . . 5 ⊢ 0 ∈ V | |
12 | 11, 8 | xpsn 7083 | . . . 4 ⊢ ({0} × {(𝐹‘𝑆)}) = {〈0, (𝐹‘𝑆)〉} |
13 | 10, 12 | eqtr4i 2768 | . . 3 ⊢ 〈“(𝐹‘𝑆)”〉 = ({0} × {(𝐹‘𝑆)}) |
14 | ffn 6665 | . . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
15 | id 22 | . . . 4 ⊢ (𝑆 ∈ 𝐴 → 𝑆 ∈ 𝐴) | |
16 | fcoconst 7076 | . . . 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 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 {csn 4584 〈cop 4590 × cxp 5629 ∘ ccom 5635 Fn wfn 6488 ⟶wf 6489 ‘cfv 6493 ℂcc 11007 0cc0 11009 〈“cs1 14436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-mulcl 11071 ax-i2m1 11077 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-s1 14437 |
This theorem is referenced by: cats1co 14702 s2co 14766 frmdgsum 18631 frmdup2 18634 efginvrel2 19467 vrgpinv 19509 frgpup2 19516 mrsubcv 33907 |
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