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Theorem mapsnconst 8911
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 5433 . . . 4 {𝑋} ∈ V
31, 2elmap 8890 . . 3 (𝐹 ∈ (𝐵m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵)
4 mapsncnv.x . . . . . 6 𝑋 ∈ V
54fsn2 7145 . . . . 5 (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹𝑋) ∈ 𝐵𝐹 = {⟨𝑋, (𝐹𝑋)⟩}))
65simprbi 495 . . . 4 (𝐹:{𝑋}⟶𝐵𝐹 = {⟨𝑋, (𝐹𝑋)⟩})
7 mapsncnv.s . . . . . 6 𝑆 = {𝑋}
87xpeq1i 5704 . . . . 5 (𝑆 × {(𝐹𝑋)}) = ({𝑋} × {(𝐹𝑋)})
9 fvex 6909 . . . . . 6 (𝐹𝑋) ∈ V
104, 9xpsn 7150 . . . . 5 ({𝑋} × {(𝐹𝑋)}) = {⟨𝑋, (𝐹𝑋)⟩}
118, 10eqtr2i 2754 . . . 4 {⟨𝑋, (𝐹𝑋)⟩} = (𝑆 × {(𝐹𝑋)})
126, 11eqtrdi 2781 . . 3 (𝐹:{𝑋}⟶𝐵𝐹 = (𝑆 × {(𝐹𝑋)}))
133, 12sylbi 216 . 2 (𝐹 ∈ (𝐵m {𝑋}) → 𝐹 = (𝑆 × {(𝐹𝑋)}))
147oveq2i 7430 . 2 (𝐵m 𝑆) = (𝐵m {𝑋})
1513, 14eleq2s 2843 1 (𝐹 ∈ (𝐵m 𝑆) → 𝐹 = (𝑆 × {(𝐹𝑋)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3461  {csn 4630  cop 4636   × cxp 5676  wf 6545  cfv 6549  (class class class)co 7419  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8847
This theorem is referenced by:  mapsncnv  8912  fvcoe1  22150  coe1mul2lem1  22211  coe1mul2  22213  0prjspnrel  42186
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