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Theorem mapsnconst 8458

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 5334	. . . 4 ⊢ {𝑋} ∈ V
3	1, 2	elmap 8437	. . 3 ⊢ (𝐹 ∈ (𝐵 ↑_m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵)
4		mapsncnv.x	. . . . . 6 ⊢ 𝑋 ∈ V
5	4	fsn2 6900	. . . . 5 ⊢ (𝐹:{𝑋}⟶𝐵 ↔ ((𝐹‘𝑋) ∈ 𝐵 ∧ 𝐹 = {⟨𝑋, (𝐹‘𝑋)⟩}))
6	5	simprbi 499	. . . 4 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = {⟨𝑋, (𝐹‘𝑋)⟩})
7		mapsncnv.s	. . . . . 6 ⊢ 𝑆 = {𝑋}
8	7	xpeq1i 5583	. . . . 5 ⊢ (𝑆 × {(𝐹‘𝑋)}) = ({𝑋} × {(𝐹‘𝑋)})
9		fvex 6685	. . . . . 6 ⊢ (𝐹‘𝑋) ∈ V
10	4, 9	xpsn 6905	. . . . 5 ⊢ ({𝑋} × {(𝐹‘𝑋)}) = {⟨𝑋, (𝐹‘𝑋)⟩}
11	8, 10	eqtr2i 2847	. . . 4 ⊢ {⟨𝑋, (𝐹‘𝑋)⟩} = (𝑆 × {(𝐹‘𝑋)})
12	6, 11	syl6eq 2874	. . 3 ⊢ (𝐹:{𝑋}⟶𝐵 → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))
13	3, 12	sylbi 219	. 2 ⊢ (𝐹 ∈ (𝐵 ↑_m {𝑋}) → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))
14	7	oveq2i 7169	. 2 ⊢ (𝐵 ↑_m 𝑆) = (𝐵 ↑_m {𝑋})
15	13, 14	eleq2s 2933	1 ⊢ (𝐹 ∈ (𝐵 ↑_m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 {csn 4569 ⟨cop 4575 × cxp 5555 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ↑_m cmap 8408

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410

This theorem is referenced by: mapsncnv 8459 fvcoe1 20377 coe1mul2lem1 20437 coe1mul2 20439 0prjspnrel 39276

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