Theorem mapsnconst 8638

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 ∈ (𝐵 ↑m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))

Proof of Theorem mapsnconst

Step	Hyp	Ref	Expression
1		mapsncnv.b	. . . 4 ⊢ 𝐵 ∈ V
2		snex 5349	. . . 4 ⊢ {𝑋} ∈ V
3	1, 2	elmap 8617	. . 3 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵)
4		mapsncnv.x	. . . . . 6 ⊢ 𝑋 ∈ V
5	4	fsn2 6990	. . . . 5 ⊢ (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹‘𝑋) ∈ 𝐵 ∧ 𝐹 = {⟨𝑋, (𝐹‘𝑋)⟩}))
6	5	simprbi 496	. . . 4 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = {⟨𝑋, (𝐹‘𝑋)⟩})
7		mapsncnv.s	. . . . . 6 ⊢ 𝑆 = {𝑋}
8	7	xpeq1i 5606	. . . . 5 ⊢ (𝑆 × {(𝐹‘𝑋)}) = ({𝑋} × {(𝐹‘𝑋)})
9		fvex 6769	. . . . . 6 ⊢ (𝐹‘𝑋) ∈ V
10	4, 9	xpsn 6995	. . . . 5 ⊢ ({𝑋} × {(𝐹‘𝑋)}) = {⟨𝑋, (𝐹‘𝑋)⟩}
11	8, 10	eqtr2i 2767	. . . 4 ⊢ {⟨𝑋, (𝐹‘𝑋)⟩} = (𝑆 × {(𝐹‘𝑋)})
12	6, 11	eqtrdi 2795	. . 3 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))
13	3, 12	sylbi 216	. 2 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))
14	7	oveq2i 7266	. 2 ⊢ (𝐵 ↑m 𝑆) = (𝐵 ↑m {𝑋})
15	13, 14	eleq2s 2857	1 ⊢ (𝐹 ∈ (𝐵 ↑m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {csn 4558  ⟨cop 4564   × cxp 5578  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575

This theorem is referenced by:  mapsncnv  8639  fvcoe1  21288  coe1mul2lem1  21348  coe1mul2  21350  0prjspnrel  40385