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Theorem mapsnconst 8931
Description: Every singleton map is a constant function. (Contributed by Stefan O'Rear, 25-Mar-2015.)
Hypotheses
Ref Expression
mapsncnv.s 𝑆 = {𝑋}
mapsncnv.b 𝐵 ∈ V
mapsncnv.x 𝑋 ∈ V
Assertion
Ref Expression
mapsnconst (𝐹 ∈ (𝐵m 𝑆) → 𝐹 = (𝑆 × {(𝐹𝑋)}))

Proof of Theorem mapsnconst
StepHypRef Expression
1 mapsncnv.b . . . 4 𝐵 ∈ V
2 snex 5442 . . . 4 {𝑋} ∈ V
31, 2elmap 8910 . . 3 (𝐹 ∈ (𝐵m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵)
4 mapsncnv.x . . . . . 6 𝑋 ∈ V
54fsn2 7156 . . . . 5 (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹𝑋) ∈ 𝐵𝐹 = {⟨𝑋, (𝐹𝑋)⟩}))
65simprbi 496 . . . 4 (𝐹:{𝑋}⟶𝐵𝐹 = {⟨𝑋, (𝐹𝑋)⟩})
7 mapsncnv.s . . . . . 6 𝑆 = {𝑋}
87xpeq1i 5715 . . . . 5 (𝑆 × {(𝐹𝑋)}) = ({𝑋} × {(𝐹𝑋)})
9 fvex 6920 . . . . . 6 (𝐹𝑋) ∈ V
104, 9xpsn 7161 . . . . 5 ({𝑋} × {(𝐹𝑋)}) = {⟨𝑋, (𝐹𝑋)⟩}
118, 10eqtr2i 2764 . . . 4 {⟨𝑋, (𝐹𝑋)⟩} = (𝑆 × {(𝐹𝑋)})
126, 11eqtrdi 2791 . . 3 (𝐹:{𝑋}⟶𝐵𝐹 = (𝑆 × {(𝐹𝑋)}))
133, 12sylbi 217 . 2 (𝐹 ∈ (𝐵m {𝑋}) → 𝐹 = (𝑆 × {(𝐹𝑋)}))
147oveq2i 7442 . 2 (𝐵m 𝑆) = (𝐵m {𝑋})
1513, 14eleq2s 2857 1 (𝐹 ∈ (𝐵m 𝑆) → 𝐹 = (𝑆 × {(𝐹𝑋)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631  cop 4637   × cxp 5687  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865
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-pow 5371  ax-pr 5438  ax-un 7754
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-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  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-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867
This theorem is referenced by:  mapsncnv  8932  fvcoe1  22225  coe1mul2lem1  22286  coe1mul2  22288  0prjspnrel  42614
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