Theorem mapsncnv 8912

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 8868	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → 𝑥:{𝑋}⟶𝐵)
2		mapsncnv.x	. . . . . . . . . 10 ⊢ 𝑋 ∈ V
3	2	snid 4643	. . . . . . . . 9 ⊢ 𝑋 ∈ {𝑋}
4		ffvelcdm 7076	. . . . . . . . 9 ⊢ ((𝑥:{𝑋}⟶𝐵 ∧ 𝑋 ∈ {𝑋}) → (𝑥‘𝑋) ∈ 𝐵)
5	1, 3, 4	sylancl 586	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑m {𝑋}) → (𝑥‘𝑋) ∈ 𝐵)
6		eqid 2736	. . . . . . . . 9 ⊢ {𝑋} = {𝑋}
7		mapsncnv.b	. . . . . . . . 9 ⊢ 𝐵 ∈ V
8	6, 7, 2	mapsnconst 8911	. . . . . . . 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 2823	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝑋) → (𝑦 ∈ 𝐵 ↔ (𝑥‘𝑋) ∈ 𝐵))
11		sneq 4616	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥‘𝑋) → {𝑦} = {(𝑥‘𝑋)})
12	11	xpeq2d 5689	. . . . . . . . 9 ⊢ (𝑦 = (𝑥‘𝑋) → ({𝑋} × {𝑦}) = ({𝑋} × {(𝑥‘𝑋)}))
13	12	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝑋) → (𝑥 = ({𝑋} × {𝑦}) ↔ 𝑥 = ({𝑋} × {(𝑥‘𝑋)})))
14	10, 13	anbi12d 632	. . . . . . 7 ⊢ (𝑦 = (𝑥‘𝑋) → ((𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})) ↔ ((𝑥‘𝑋) ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {(𝑥‘𝑋)}))))
15	9, 14	syl5ibrcom 247	. . . . . 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 6772	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
18		snex 5411	. . . . . . . . . 10 ⊢ {𝑋} ∈ V
19	7, 18	elmap 8890	. . . . . . . . 9 ⊢ (({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋}) ↔ ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
20	17, 19	sylibr 234	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋}))
21		vex 3468	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	fvconst2 7201	. . . . . . . . . 10 ⊢ (𝑋 ∈ {𝑋} → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
23	3, 22	mp1i 13	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
24	23	eqcomd 2742	. . . . . . . 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 2823	. . . . . . . 8 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵 ↑m {𝑋}) ↔ ({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋})))
27		fveq1 6880	. . . . . . . . 9 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑥‘𝑋) = (({𝑋} × {𝑦})‘𝑋))
28	27	eqeq2d 2747	. . . . . . . 8 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑦 = (𝑥‘𝑋) ↔ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
29	26, 28	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = ({𝑋} × {𝑦}) → ((𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (({𝑋} × {𝑦}) ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋))))
30	25, 29	syl5ibrcom 247	. . . . . 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 209	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
33		mapsncnv.s	. . . . . . 7 ⊢ 𝑆 = {𝑋}
34	33	oveq2i 7421	. . . . . 6 ⊢ (𝐵 ↑m 𝑆) = (𝐵 ↑m {𝑋})
35	34	eleq2i 2827	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m 𝑆) ↔ 𝑥 ∈ (𝐵 ↑m {𝑋}))
36	35	anbi1i 624	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑥 ∈ (𝐵 ↑m {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)))
37	33	xpeq1i 5685	. . . . . 6 ⊢ (𝑆 × {𝑦}) = ({𝑋} × {𝑦})
38	37	eqeq2i 2749	. . . . 5 ⊢ (𝑥 = (𝑆 × {𝑦}) ↔ 𝑥 = ({𝑋} × {𝑦}))
39	38	anbi2i 623	. . . 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 5191	. 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 5207	. . . . 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 5859	. . 3 ⊢ ◡𝐹 = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑m 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
46		cnvopab 6131	. . 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 5207	. 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 2109  Vcvv 3464  {csn 4606  {copab 5186   ↦ cmpt 5206   × cxp 5657  ^◡ccnv 5658  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-map 8847

This theorem is referenced by:  mapsnf1o2  8913  mapsnf1o3  8914  coe1sfi  22154  evl1var  22279  pf1mpf  22295  pf1ind  22298  deg1val  26058