|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > fconstg | Structured version Visualization version GIF version | ||
| Description: A Cartesian product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.) | 
| Ref | Expression | 
|---|---|
| fconstg | ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | sneq 4635 | . . . 4 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵}) | |
| 2 | 1 | xpeq2d 5714 | . . 3 ⊢ (𝑥 = 𝐵 → (𝐴 × {𝑥}) = (𝐴 × {𝐵})) | 
| 3 | feq1 6715 | . . . 4 ⊢ ((𝐴 × {𝑥}) = (𝐴 × {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝑥})) | |
| 4 | feq3 6717 | . . . 4 ⊢ ({𝑥} = {𝐵} → ((𝐴 × {𝐵}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) | |
| 5 | 3, 4 | sylan9bb 509 | . . 3 ⊢ (((𝐴 × {𝑥}) = (𝐴 × {𝐵}) ∧ {𝑥} = {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) | 
| 6 | 2, 1, 5 | syl2anc 584 | . 2 ⊢ (𝑥 = 𝐵 → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) | 
| 7 | vex 3483 | . . 3 ⊢ 𝑥 ∈ V | |
| 8 | 7 | fconst 6793 | . 2 ⊢ (𝐴 × {𝑥}):𝐴⟶{𝑥} | 
| 9 | 6, 8 | vtoclg 3553 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 {csn 4625 × cxp 5682 ⟶wf 6556 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-fun 6562 df-fn 6563 df-f 6564 | 
| This theorem is referenced by: fnconstg 6795 fconst6g 6796 xpsng 7158 fvconst2g 7223 fconst2g 7224 symgpssefmnd 19414 xkoptsub 23663 mbfconstlem 25663 i1fmulclem 25738 i1fmulc 25739 itg2mulclem 25782 dvcmulf 25983 dvef 26019 coemulc 26295 resf1o 32742 locfinref 33841 ccatmulgnn0dir 34558 frlmvscadiccat 42521 | 
| Copyright terms: Public domain | W3C validator |