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Theorem mapsnconst 8448
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 5320 . . . 4 {𝑋} ∈ V
31, 2elmap 8427 . . 3 (𝐹 ∈ (𝐵m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵)
4 mapsncnv.x . . . . . 6 𝑋 ∈ V
54fsn2 6887 . . . . 5 (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹𝑋) ∈ 𝐵𝐹 = {⟨𝑋, (𝐹𝑋)⟩}))
65simprbi 500 . . . 4 (𝐹:{𝑋}⟶𝐵𝐹 = {⟨𝑋, (𝐹𝑋)⟩})
7 mapsncnv.s . . . . . 6 𝑆 = {𝑋}
87xpeq1i 5569 . . . . 5 (𝑆 × {(𝐹𝑋)}) = ({𝑋} × {(𝐹𝑋)})
9 fvex 6672 . . . . . 6 (𝐹𝑋) ∈ V
104, 9xpsn 6892 . . . . 5 ({𝑋} × {(𝐹𝑋)}) = {⟨𝑋, (𝐹𝑋)⟩}
118, 10eqtr2i 2848 . . . 4 {⟨𝑋, (𝐹𝑋)⟩} = (𝑆 × {(𝐹𝑋)})
126, 11syl6eq 2875 . . 3 (𝐹:{𝑋}⟶𝐵𝐹 = (𝑆 × {(𝐹𝑋)}))
133, 12sylbi 220 . 2 (𝐹 ∈ (𝐵m {𝑋}) → 𝐹 = (𝑆 × {(𝐹𝑋)}))
147oveq2i 7157 . 2 (𝐵m 𝑆) = (𝐵m {𝑋})
1513, 14eleq2s 2934 1 (𝐹 ∈ (𝐵m 𝑆) → 𝐹 = (𝑆 × {(𝐹𝑋)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  Vcvv 3480  {csn 4550  cop 4556   × cxp 5541  wf 6340  cfv 6344  (class class class)co 7146  m cmap 8398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-map 8400
This theorem is referenced by:  mapsncnv  8449  fvcoe1  20370  coe1mul2lem1  20430  coe1mul2  20432  0prjspnrel  39469
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