Theorem mapsncnv 8891

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 8847	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → 𝑥:{𝑋}⟶𝐵)
2		mapsncnv.x	. . . . . . . . . 10 ⊢ 𝑋 ∈ V
3	2	snid 4664	. . . . . . . . 9 ⊢ 𝑋 ∈ {𝑋}
4		ffvelcdm 7083	. . . . . . . . 9 ⊢ ((𝑥:{𝑋}⟶𝐵 ∧ 𝑋 ∈ {𝑋}) → (𝑥‘𝑋) ∈ 𝐵)
5	1, 3, 4	sylancl 585	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → (𝑥‘𝑋) ∈ 𝐵)
6		eqid 2731	. . . . . . . . 9 ⊢ {𝑋} = {𝑋}
7		mapsncnv.b	. . . . . . . . 9 ⊢ 𝐵 ∈ V
8	6, 7, 2	mapsnconst 8890	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → 𝑥 = ({𝑋} × {(𝑥‘𝑋)}))
9	5, 8	jca 511	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → ((𝑥‘𝑋) ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {(𝑥‘𝑋)})))
10		eleq1 2820	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝑋) → (𝑦 ∈ 𝐵 ↔ (𝑥‘𝑋) ∈ 𝐵))
11		sneq 4638	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥‘𝑋) → {𝑦} = {(𝑥‘𝑋)})
12	11	xpeq2d 5706	. . . . . . . . 9 ⊢ (𝑦 = (𝑥‘𝑋) → ({𝑋} × {𝑦}) = ({𝑋} × {(𝑥‘𝑋)}))
13	12	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝑋) → (𝑥 = ({𝑋} × {𝑦}) ↔ 𝑥 = ({𝑋} × {(𝑥‘𝑋)})))
14	10, 13	anbi12d 630	. . . . . . 7 ⊢ (𝑦 = (𝑥‘𝑋) → ((𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})) ↔ ((𝑥‘𝑋) ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {(𝑥‘𝑋)}))))
15	9, 14	syl5ibrcom 246	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → (𝑦 = (𝑥‘𝑋) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦}))))
16	15	imp 406	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
17		fconst6g 6780	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
18		snex 5431	. . . . . . . . . 10 ⊢ {𝑋} ∈ V
19	7, 18	elmap 8869	. . . . . . . . 9 ⊢ (({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋}) ↔ ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
20	17, 19	sylibr 233	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋}))
21		vex 3477	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	fvconst2 7207	. . . . . . . . . 10 ⊢ (𝑋 ∈ {𝑋} → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
23	3, 22	mp1i 13	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
24	23	eqcomd 2737	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → 𝑦 = (({𝑋} × {𝑦})‘𝑋))
25	20, 24	jca 511	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
26		eleq1 2820	. . . . . . . 8 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵 ↑m {𝑋}) ↔ ({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋})))
27		fveq1 6890	. . . . . . . . 9 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑥‘𝑋) = (({𝑋} × {𝑦})‘𝑋))
28	27	eqeq2d 2742	. . . . . . . 8 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑦 = (𝑥‘𝑋) ↔ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
29	26, 28	anbi12d 630	. . . . . . 7 ⊢ (𝑥 = ({𝑋} × {𝑦}) → ((𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋))))
30	25, 29	syl5ibrcom 246	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋))))
31	30	imp 406	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})) → (𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)))
32	16, 31	impbii 208	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
33		mapsncnv.s	. . . . . . 7 ⊢ 𝑆 = {𝑋}
34	33	oveq2i 7423	. . . . . 6 ⊢ (𝐵 ↑m 𝑆) = (𝐵 ↑m {𝑋})
35	34	eleq2i 2824	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↔ 𝑥 ∈ (𝐵 ↑m {𝑋}))
36	35	anbi1i 623	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)))
37	33	xpeq1i 5702	. . . . . 6 ⊢ (𝑆 × {𝑦}) = ({𝑋} × {𝑦})
38	37	eqeq2i 2744	. . . . 5 ⊢ (𝑥 = (𝑆 × {𝑦}) ↔ 𝑥 = ({𝑋} × {𝑦}))
39	38	anbi2i 622	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦})) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
40	32, 36, 39	3bitr4i 303	. . 3 ⊢ ((𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦})))
41	40	opabbii 5215	. 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 5232	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↦ (𝑥‘𝑋)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
44	42, 43	eqtri 2759	. . . 4 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
45	44	cnveqi 5874	. . 3 ⊢ ◡𝐹 = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
46		cnvopab 6138	. . 3 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
47	45, 46	eqtri 2759	. 2 ⊢ ◡𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
48		df-mpt 5232	. 2 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦}))}
49	41, 47, 48	3eqtr4i 2769	1 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2105  Vcvv 3473  {csn 4628  {copab 5210   ↦ cmpt 5231   × cxp 5674  ^◡ccnv 5675  ⟶wf 6539  ‘cfv 6543  (class class class)co 7412   ↑_m cmap 8824

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 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-map 8826

This theorem is referenced by:  mapsnf1o2  8892  mapsnf1o3  8893  coe1sfi  21957  evl1var  22076  pf1mpf  22092  pf1ind  22095  deg1val  25850