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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 4641 | . . . 4 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵}) | |
2 | 1 | xpeq2d 5719 | . . 3 ⊢ (𝑥 = 𝐵 → (𝐴 × {𝑥}) = (𝐴 × {𝐵})) |
3 | feq1 6717 | . . . 4 ⊢ ((𝐴 × {𝑥}) = (𝐴 × {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝑥})) | |
4 | feq3 6719 | . . . 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 3482 | . . 3 ⊢ 𝑥 ∈ V | |
8 | 7 | fconst 6795 | . 2 ⊢ (𝐴 × {𝑥}):𝐴⟶{𝑥} |
9 | 6, 8 | vtoclg 3554 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 {csn 4631 × cxp 5687 ⟶wf 6559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-fun 6565 df-fn 6566 df-f 6567 |
This theorem is referenced by: fnconstg 6797 fconst6g 6798 xpsng 7159 fvconst2g 7222 fconst2g 7223 symgpssefmnd 19428 xkoptsub 23678 mbfconstlem 25676 i1fmulclem 25752 i1fmulc 25753 itg2mulclem 25796 dvcmulf 25997 dvef 26033 coemulc 26309 resf1o 32748 locfinref 33802 ccatmulgnn0dir 34536 frlmvscadiccat 42493 |
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