Theorem mapsnconst 8743

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 5373	. . . 4 ⊢ {𝑋} ∈ V
3	1, 2	elmap 8722	. . 3 ⊢ (𝐹 ∈ (𝐵 ↑m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵)
4		mapsncnv.x	. . . . . 6 ⊢ 𝑋 ∈ V
5	4	fsn2 7058	. . . . 5 ⊢ (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹‘𝑋) ∈ 𝐵 ∧ 𝐹 = {⟨𝑋, (𝐹‘𝑋)⟩}))
6	5	simprbi 497	. . . 4 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = {⟨𝑋, (𝐹‘𝑋)⟩})
7		mapsncnv.s	. . . . . 6 ⊢ 𝑆 = {𝑋}
8	7	xpeq1i 5640	. . . . 5 ⊢ (𝑆 × {(𝐹‘𝑋)}) = ({𝑋} × {(𝐹‘𝑋)})
9		fvex 6832	. . . . . 6 ⊢ (𝐹‘𝑋) ∈ V
10	4, 9	xpsn 7063	. . . . 5 ⊢ ({𝑋} × {(𝐹‘𝑋)}) = {⟨𝑋, (𝐹‘𝑋)⟩}
11	8, 10	eqtr2i 2765	. . . 4 ⊢ {⟨𝑋, (𝐹‘𝑋)⟩} = (𝑆 × {(𝐹‘𝑋)})
12	6, 11	eqtrdi 2792	. . 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 7340	. 2 ⊢ (𝐵 ↑m 𝑆) = (𝐵 ↑m {𝑋})
15	13, 14	eleq2s 2855	1 ⊢ (𝐹 ∈ (𝐵 ↑m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  Vcvv 3441  {csn 4572  ⟨cop 4578   × cxp 5612  ⟶wf 6469  ‘cfv 6473  (class class class)co 7329   ↑_m cmap 8678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-ov 7332  df-oprab 7333  df-mpo 7334  df-map 8680

This theorem is referenced by:  mapsncnv  8744  fvcoe1  21476  coe1mul2lem1  21536  coe1mul2  21538  0prjspnrel  40714