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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 4579 | . . . 4 ⊢ (𝑥 = 𝐵 → {𝑥} = {𝐵}) | |
2 | 1 | xpeq2d 5587 | . . 3 ⊢ (𝑥 = 𝐵 → (𝐴 × {𝑥}) = (𝐴 × {𝐵})) |
3 | feq1 6497 | . . . 4 ⊢ ((𝐴 × {𝑥}) = (𝐴 × {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝑥})) | |
4 | feq3 6499 | . . . 4 ⊢ ({𝑥} = {𝐵} → ((𝐴 × {𝐵}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) | |
5 | 3, 4 | sylan9bb 512 | . . 3 ⊢ (((𝐴 × {𝑥}) = (𝐴 × {𝐵}) ∧ {𝑥} = {𝐵}) → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) |
6 | 2, 1, 5 | syl2anc 586 | . 2 ⊢ (𝑥 = 𝐵 → ((𝐴 × {𝑥}):𝐴⟶{𝑥} ↔ (𝐴 × {𝐵}):𝐴⟶{𝐵})) |
7 | vex 3499 | . . 3 ⊢ 𝑥 ∈ V | |
8 | 7 | fconst 6567 | . 2 ⊢ (𝐴 × {𝑥}):𝐴⟶{𝑥} |
9 | 6, 8 | vtoclg 3569 | 1 ⊢ (𝐵 ∈ 𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 {csn 4569 × cxp 5555 ⟶wf 6353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-fun 6359 df-fn 6360 df-f 6361 |
This theorem is referenced by: fnconstg 6569 fconst6g 6570 xpsng 6903 fvconst2g 6966 fconst2g 6967 symgpssefmnd 18526 xkoptsub 22264 mbfconstlem 24230 i1fmulclem 24305 i1fmulc 24306 itg2mulclem 24349 dvcmulf 24544 dvef 24579 coemulc 24847 resf1o 30468 locfinref 31107 ccatmulgnn0dir 31814 frlmvscadiccat 39152 |
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