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Theorem mapsnconst 8932
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 5436 . . . 4 {𝑋} ∈ V
31, 2elmap 8911 . . 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 5711 . . . . 5 (𝑆 × {(𝐹𝑋)}) = ({𝑋} × {(𝐹𝑋)})
9 fvex 6919 . . . . . 6 (𝐹𝑋) ∈ V
104, 9xpsn 7161 . . . . 5 ({𝑋} × {(𝐹𝑋)}) = {⟨𝑋, (𝐹𝑋)⟩}
118, 10eqtr2i 2766 . . . 4 {⟨𝑋, (𝐹𝑋)⟩} = (𝑆 × {(𝐹𝑋)})
126, 11eqtrdi 2793 . . 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 2859 1 (𝐹 ∈ (𝐵m 𝑆) → 𝐹 = (𝑆 × {(𝐹𝑋)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632   × cxp 5683  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868
This theorem is referenced by:  mapsncnv  8933  fvcoe1  22209  coe1mul2lem1  22270  coe1mul2  22272  0prjspnrel  42637
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