Theorem mapsncnv 8192

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 8164	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) → 𝑥:{𝑋}⟶𝐵)
2		mapsncnv.x	. . . . . . . . . 10 ⊢ 𝑋 ∈ V
3	2	snid 4430	. . . . . . . . 9 ⊢ 𝑋 ∈ {𝑋}
4		ffvelrn 6623	. . . . . . . . 9 ⊢ ((𝑥:{𝑋}⟶𝐵 ∧ 𝑋 ∈ {𝑋}) → (𝑥‘𝑋) ∈ 𝐵)
5	1, 3, 4	sylancl 580	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) → (𝑥‘𝑋) ∈ 𝐵)
6		eqid 2778	. . . . . . . . 9 ⊢ {𝑋} = {𝑋}
7		mapsncnv.b	. . . . . . . . 9 ⊢ 𝐵 ∈ V
8	6, 7, 2	mapsnconst 8191	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) → 𝑥 = ({𝑋} × {(𝑥‘𝑋)}))
9	5, 8	jca 507	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) → ((𝑥‘𝑋) ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {(𝑥‘𝑋)})))
10		eleq1 2847	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝑋) → (𝑦 ∈ 𝐵 ↔ (𝑥‘𝑋) ∈ 𝐵))
11		sneq 4408	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥‘𝑋) → {𝑦} = {(𝑥‘𝑋)})
12	11	xpeq2d 5387	. . . . . . . . 9 ⊢ (𝑦 = (𝑥‘𝑋) → ({𝑋} × {𝑦}) = ({𝑋} × {(𝑥‘𝑋)}))
13	12	eqeq2d 2788	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝑋) → (𝑥 = ({𝑋} × {𝑦}) ↔ 𝑥 = ({𝑋} × {(𝑥‘𝑋)})))
14	10, 13	anbi12d 624	. . . . . . 7 ⊢ (𝑦 = (𝑥‘𝑋) → ((𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})) ↔ ((𝑥‘𝑋) ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {(𝑥‘𝑋)}))))
15	9, 14	syl5ibrcom 239	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) → (𝑦 = (𝑥‘𝑋) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦}))))
16	15	imp 397	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
17		fconst6g 6346	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
18		snex 5142	. . . . . . . . . 10 ⊢ {𝑋} ∈ V
19	7, 18	elmap 8171	. . . . . . . . 9 ⊢ (({𝑋} × {𝑦}) ∈ (𝐵 ↑𝑚 {𝑋}) ↔ ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
20	17, 19	sylibr 226	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ({𝑋} × {𝑦}) ∈ (𝐵 ↑𝑚 {𝑋}))
21		vex 3401	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	fvconst2 6743	. . . . . . . . . 10 ⊢ (𝑋 ∈ {𝑋} → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
23	3, 22	mp1i 13	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
24	23	eqcomd 2784	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → 𝑦 = (({𝑋} × {𝑦})‘𝑋))
25	20, 24	jca 507	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (({𝑋} × {𝑦}) ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
26		eleq1 2847	. . . . . . . 8 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ↔ ({𝑋} × {𝑦}) ∈ (𝐵 ↑𝑚 {𝑋})))
27		fveq1 6447	. . . . . . . . 9 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑥‘𝑋) = (({𝑋} × {𝑦})‘𝑋))
28	27	eqeq2d 2788	. . . . . . . 8 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑦 = (𝑥‘𝑋) ↔ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
29	26, 28	anbi12d 624	. . . . . . 7 ⊢ (𝑥 = ({𝑋} × {𝑦}) → ((𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (({𝑋} × {𝑦}) ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋))))
30	25, 29	syl5ibrcom 239	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋))))
31	30	imp 397	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})) → (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)))
32	16, 31	impbii 201	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
33		mapsncnv.s	. . . . . . 7 ⊢ 𝑆 = {𝑋}
34	33	oveq2i 6935	. . . . . 6 ⊢ (𝐵 ↑𝑚 𝑆) = (𝐵 ↑𝑚 {𝑋})
35	34	eleq2i 2851	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↔ 𝑥 ∈ (𝐵 ↑𝑚 {𝑋}))
36	35	anbi1i 617	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)))
37	33	xpeq1i 5383	. . . . . 6 ⊢ (𝑆 × {𝑦}) = ({𝑋} × {𝑦})
38	37	eqeq2i 2790	. . . . 5 ⊢ (𝑥 = (𝑆 × {𝑦}) ↔ 𝑥 = ({𝑋} × {𝑦}))
39	38	anbi2i 616	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦})) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
40	32, 36, 39	3bitr4i 295	. . 3 ⊢ ((𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦})))
41	40	opabbii 4955	. 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 4968	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
44	42, 43	eqtri 2802	. . . 4 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
45	44	cnveqi 5544	. . 3 ⊢ ◡𝐹 = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
46		cnvopab 5790	. . 3 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
47	45, 46	eqtri 2802	. 2 ⊢ ◡𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
48		df-mpt 4968	. 2 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦}))}
49	41, 47, 48	3eqtr4i 2812	1 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))

Syntax hints:   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398  {csn 4398  {copab 4950   ↦ cmpt 4967   × cxp 5355  ^◡ccnv 5356  ⟶wf 6133  ‘cfv 6137  (class class class)co 6924   ↑_𝑚 cmap 8142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-1st 7447  df-2nd 7448  df-map 8144

This theorem is referenced by:  mapsnf1o2  8193  mapsnf1o3  8194  coe1sfi  19983  evl1var  20100  pf1mpf  20116  pf1ind  20119  deg1val  24297