Theorem mapsnconst 8906

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 5406	. . . 4 ⊢ {𝑋} ∈ V
3	1, 2	elmap 8885	. . 3 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵)
4		mapsncnv.x	. . . . . 6 ⊢ 𝑋 ∈ V
5	4	fsn2 7126	. . . . 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 5680	. . . . 5 ⊢ (𝑆 × {(𝐹‘𝑋)}) = ({𝑋} × {(𝐹‘𝑋)})
9		fvex 6889	. . . . . 6 ⊢ (𝐹‘𝑋) ∈ V
10	4, 9	xpsn 7131	. . . . 5 ⊢ ({𝑋} × {(𝐹‘𝑋)}) = {⟨𝑋, (𝐹‘𝑋)⟩}
11	8, 10	eqtr2i 2759	. . . 4 ⊢ {⟨𝑋, (𝐹‘𝑋)⟩} = (𝑆 × {(𝐹‘𝑋)})
12	6, 11	eqtrdi 2786	. . 3 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))
13	3, 12	sylbi 217	. 2 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))
14	7	oveq2i 7416	. 2 ⊢ (𝐵 ↑m 𝑆) = (𝐵 ↑m {𝑋})
15	13, 14	eleq2s 2852	1 ⊢ (𝐹 ∈ (𝐵 ↑m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  Vcvv 3459  {csn 4601  ⟨cop 4607   × cxp 5652  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405   ↑_m cmap 8840

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 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8842

This theorem is referenced by:  mapsncnv  8907  fvcoe1  22143  coe1mul2lem1  22204  coe1mul2  22206  0prjspnrel  42650