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Theorem mapsnconst 8456

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 5332	. . . 4 ⊢ {𝑋} ∈ V
3	1, 2	elmap 8435	. . 3 ⊢ (𝐹 ∈ (𝐵 ↑_m {𝑋}) ↔ 𝐹:{𝑋}⟶𝐵)
4		mapsncnv.x	. . . . . 6 ⊢ 𝑋 ∈ V
5	4	fsn2 6898	. . . . 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 5581	. . . . 5 ⊢ (𝑆 × {(𝐹‘𝑋)}) = ({𝑋} × {(𝐹‘𝑋)})
9		fvex 6683	. . . . . 6 ⊢ (𝐹‘𝑋) ∈ V
10	4, 9	xpsn 6903	. . . . 5 ⊢ ({𝑋} × {(𝐹‘𝑋)}) = {⟨𝑋, (𝐹‘𝑋)⟩}
11	8, 10	eqtr2i 2845	. . . 4 ⊢ {⟨𝑋, (𝐹‘𝑋)⟩} = (𝑆 × {(𝐹‘𝑋)})
12	6, 11	syl6eq 2872	. . 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 7167	. 2 ⊢ (𝐵 ↑_m 𝑆) = (𝐵 ↑_m {𝑋})
15	13, 14	eleq2s 2931	1 ⊢ (𝐹 ∈ (𝐵 ↑_m 𝑆) → 𝐹 = (𝑆 × {(𝐹‘𝑋)}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3494 {csn 4567 ⟨cop 4573 × cxp 5553 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑_m cmap 8406

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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461

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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-map 8408

This theorem is referenced by: mapsncnv 8457 fvcoe1 20375 coe1mul2lem1 20435 coe1mul2 20437 0prjspnrel 39289

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