Theorem mapsncnv 8894

Description: Expression for the inverse of the canonical map between a set and its set of singleton functions. (Contributed by Stefan O'Rear, 21-Mar-2015.)

Hypotheses

Ref	Expression
mapsncnv.s	⊢ 𝑆 = {𝑋}
mapsncnv.b	⊢ 𝐵 ∈ V
mapsncnv.x	⊢ 𝑋 ∈ V
mapsncnv.f	⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋))

Assertion

Ref	Expression
mapsncnv	⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑆,𝑦   𝑦,𝑋

Allowed substitution hints:   𝐹(𝑥,𝑦)   𝑋(𝑥)

Proof of Theorem mapsncnv

Step	Hyp	Ref	Expression
1		elmapi 8849	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → 𝑥:{𝑋}⟶𝐵)
2		mapsncnv.x	. . . . . . . . . 10 ⊢ 𝑋 ∈ V
3	2	snid 4633	. . . . . . . . 9 ⊢ 𝑋 ∈ {𝑋}
4		ffvelcdm 7080	. . . . . . . . 9 ⊢ ((𝑥:{𝑋}⟶𝐵 ∧ 𝑋 ∈ {𝑋}) → (𝑥‘𝑋) ∈ 𝐵)
5	1, 3, 4	sylancl 597	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → (𝑥‘𝑋) ∈ 𝐵)
6		eqid 2770	. . . . . . . . 9 ⊢ {𝑋} = {𝑋}
7		mapsncnv.b	. . . . . . . . 9 ⊢ 𝐵 ∈ V
8	6, 7, 2	mapsnconst 8893	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → 𝑥 = ({𝑋} × {(𝑥‘𝑋)}))
9	5, 8	jca 520	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → ((𝑥‘𝑋) ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {(𝑥‘𝑋)})))
10		eleq1 2858	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝑋) → (𝑦 ∈ 𝐵 ↔ (𝑥‘𝑋) ∈ 𝐵))
11		sneq 4604	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥‘𝑋) → {𝑦} = {(𝑥‘𝑋)})
12	11	xpeq2d 5695	. . . . . . . . 9 ⊢ (𝑦 = (𝑥‘𝑋) → ({𝑋} × {𝑦}) = ({𝑋} × {(𝑥‘𝑋)}))
13	12	eqeq2d 2781	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝑋) → (𝑥 = ({𝑋} × {𝑦}) ↔ 𝑥 = ({𝑋} × {(𝑥‘𝑋)})))
14	10, 13	anbi12d 643	. . . . . . 7 ⊢ (𝑦 = (𝑥‘𝑋) → ((𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})) ↔ ((𝑥‘𝑋) ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {(𝑥‘𝑋)}))))
15	9, 14	syl5ibrcom 250	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → (𝑦 = (𝑥‘𝑋) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦}))))
16	15	imp 411	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
17		fconst6g 6771	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
18		snex 5414	. . . . . . . . . 10 ⊢ {𝑋} ∈ V
19	7, 18	elmap 8872	. . . . . . . . 9 ⊢ (({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋}) ↔ ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
20	17, 19	sylibr 237	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋}))
21		vex 3466	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	fvconst2 7206	. . . . . . . . . 10 ⊢ (𝑋 ∈ {𝑋} → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
23	3, 22	mp1i 14	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
24	23	eqcomd 2776	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → 𝑦 = (({𝑋} × {𝑦})‘𝑋))
25	20, 24	jca 520	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
26		eleq1 2858	. . . . . . . 8 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵 ↑m {𝑋}) ↔ ({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋})))
27		fveq1 6884	. . . . . . . . 9 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑥‘𝑋) = (({𝑋} × {𝑦})‘𝑋))
28	27	eqeq2d 2781	. . . . . . . 8 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑦 = (𝑥‘𝑋) ↔ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
29	26, 28	anbi12d 643	. . . . . . 7 ⊢ (𝑥 = ({𝑋} × {𝑦}) → ((𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋))))
30	25, 29	syl5ibrcom 250	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋))))
31	30	imp 411	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})) → (𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)))
32	16, 31	impbii 212	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
33		mapsncnv.s	. . . . . . 7 ⊢ 𝑆 = {𝑋}
34	33	oveq2i 7425	. . . . . 6 ⊢ (𝐵 ↑m 𝑆) = (𝐵 ↑m {𝑋})
35	34	eleq2i 2862	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↔ 𝑥 ∈ (𝐵 ↑m {𝑋}))
36	35	anbi1i 635	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)))
37	33	xpeq1i 5691	. . . . . 6 ⊢ (𝑆 × {𝑦}) = ({𝑋} × {𝑦})
38	37	eqeq2i 2783	. . . . 5 ⊢ (𝑥 = (𝑆 × {𝑦}) ↔ 𝑥 = ({𝑋} × {𝑦}))
39	38	anbi2i 634	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦})) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
40	32, 36, 39	3bitr4i 306	. . 3 ⊢ ((𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦})))
41	40	opabbii 5183	. 2 ⊢ {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))} = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦}))}
42		mapsncnv.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋))
43		df-mpt 5198	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
44	42, 43	eqtri 2793	. . . 4 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
45	44	cnveqi 5864	. . 3 ⊢ ◡𝐹 = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
46		cnvopab 6141	. . 3 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
47	45, 46	eqtri 2793	. 2 ⊢ ◡𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
48		df-mpt 5198	. 2 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦}))}
49	41, 47, 48	3eqtr4i 2803	1 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))

Syntax hints:   ∧ wa 400   = wceq 1568   ∈ wcel 2150  Vcvv 3462  {csn 4594  {copab 5178   ↦ cmpt 5197   × cxp 5663  ^◡ccnv 5664  ⟶wf 6536  ‘cfv 6540  (class class class)co 7414   ↑_m cmap 8827

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7989  df-2nd 7990  df-map 8829

This theorem is referenced by:  mapsnf1o2  8895  mapsnf1o3  8896  coe1sfi  22356  evl1var  22479  pf1mpf  22495  pf1ind  22498  deg1val  26236