Theorem mapsncnv 6697

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 6672	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) → 𝑥:{𝑋}⟶𝐵)
2		mapsncnv.x	. . . . . . . . . 10 ⊢ 𝑋 ∈ V
3	2	snid 3625	. . . . . . . . 9 ⊢ 𝑋 ∈ {𝑋}
4		ffvelcdm 5651	. . . . . . . . 9 ⊢ ((𝑥:{𝑋}⟶𝐵 ∧ 𝑋 ∈ {𝑋}) → (𝑥‘𝑋) ∈ 𝐵)
5	1, 3, 4	sylancl 413	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) → (𝑥‘𝑋) ∈ 𝐵)
6		eqid 2177	. . . . . . . . 9 ⊢ {𝑋} = {𝑋}
7		mapsncnv.b	. . . . . . . . 9 ⊢ 𝐵 ∈ V
8	6, 7, 2	mapsnconst 6696	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) → 𝑥 = ({𝑋} × {(𝑥‘𝑋)}))
9	5, 8	jca 306	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) → ((𝑥‘𝑋) ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {(𝑥‘𝑋)})))
10		eleq1 2240	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝑋) → (𝑦 ∈ 𝐵 ↔ (𝑥‘𝑋) ∈ 𝐵))
11		sneq 3605	. . . . . . . . . 10 ⊢ (𝑦 = (𝑥‘𝑋) → {𝑦} = {(𝑥‘𝑋)})
12	11	xpeq2d 4652	. . . . . . . . 9 ⊢ (𝑦 = (𝑥‘𝑋) → ({𝑋} × {𝑦}) = ({𝑋} × {(𝑥‘𝑋)}))
13	12	eqeq2d 2189	. . . . . . . 8 ⊢ (𝑦 = (𝑥‘𝑋) → (𝑥 = ({𝑋} × {𝑦}) ↔ 𝑥 = ({𝑋} × {(𝑥‘𝑋)})))
14	10, 13	anbi12d 473	. . . . . . 7 ⊢ (𝑦 = (𝑥‘𝑋) → ((𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})) ↔ ((𝑥‘𝑋) ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {(𝑥‘𝑋)}))))
15	9, 14	syl5ibrcom 157	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) → (𝑦 = (𝑥‘𝑋) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦}))))
16	15	imp 124	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) → (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
17		fconst6g 5416	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
18	2	snex 4187	. . . . . . . . . 10 ⊢ {𝑋} ∈ V
19	7, 18	elmap 6679	. . . . . . . . 9 ⊢ (({𝑋} × {𝑦}) ∈ (𝐵 ↑𝑚 {𝑋}) ↔ ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
20	17, 19	sylibr 134	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → ({𝑋} × {𝑦}) ∈ (𝐵 ↑𝑚 {𝑋}))
21		vex 2742	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
22	21	fvconst2 5734	. . . . . . . . . 10 ⊢ (𝑋 ∈ {𝑋} → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
23	3, 22	mp1i 10	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝐵 → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
24	23	eqcomd 2183	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → 𝑦 = (({𝑋} × {𝑦})‘𝑋))
25	20, 24	jca 306	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (({𝑋} × {𝑦}) ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
26		eleq1 2240	. . . . . . . 8 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ↔ ({𝑋} × {𝑦}) ∈ (𝐵 ↑𝑚 {𝑋})))
27		fveq1 5516	. . . . . . . . 9 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑥‘𝑋) = (({𝑋} × {𝑦})‘𝑋))
28	27	eqeq2d 2189	. . . . . . . 8 ⊢ (𝑥 = ({𝑋} × {𝑦}) → (𝑦 = (𝑥‘𝑋) ↔ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
29	26, 28	anbi12d 473	. . . . . . 7 ⊢ (𝑥 = ({𝑋} × {𝑦}) → ((𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (({𝑋} × {𝑦}) ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋))))
30	25, 29	syl5ibrcom 157	. . . . . 6 ⊢ (𝑦 ∈ 𝐵 → (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋))))
31	30	imp 124	. . . . 5 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})) → (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)))
32	16, 31	impbii 126	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
33		mapsncnv.s	. . . . . . 7 ⊢ 𝑆 = {𝑋}
34	33	oveq2i 5888	. . . . . 6 ⊢ (𝐵 ↑𝑚 𝑆) = (𝐵 ↑𝑚 {𝑋})
35	34	eleq2i 2244	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↔ 𝑥 ∈ (𝐵 ↑𝑚 {𝑋}))
36	35	anbi1i 458	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑥 ∈ (𝐵 ↑𝑚 {𝑋}) ∧ 𝑦 = (𝑥‘𝑋)))
37	33	xpeq1i 4648	. . . . . 6 ⊢ (𝑆 × {𝑦}) = ({𝑋} × {𝑦})
38	37	eqeq2i 2188	. . . . 5 ⊢ (𝑥 = (𝑆 × {𝑦}) ↔ 𝑥 = ({𝑋} × {𝑦}))
39	38	anbi2i 457	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦})) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = ({𝑋} × {𝑦})))
40	32, 36, 39	3bitr4i 212	. . 3 ⊢ ((𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋)) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦})))
41	40	opabbii 4072	. 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 4068	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ↦ (𝑥‘𝑋)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
44	42, 43	eqtri 2198	. . . 4 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
45	44	cnveqi 4804	. . 3 ⊢ ◡𝐹 = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
46		cnvopab 5032	. . 3 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
47	45, 46	eqtri 2198	. 2 ⊢ ◡𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵 ↑𝑚 𝑆) ∧ 𝑦 = (𝑥‘𝑋))}
48		df-mpt 4068	. 2 ⊢ (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦})) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐵 ∧ 𝑥 = (𝑆 × {𝑦}))}
49	41, 47, 48	3eqtr4i 2208	1 ⊢ ◡𝐹 = (𝑦 ∈ 𝐵 ↦ (𝑆 × {𝑦}))

Syntax hints:   ∧ wa 104   = wceq 1353   ∈ wcel 2148  Vcvv 2739  {csn 3594  {copab 4065   ↦ cmpt 4066   × cxp 4626  ^◡ccnv 4627  ⟶wf 5214  ‘cfv 5218  (class class class)co 5877   ↑_𝑚 cmap 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-map 6652

This theorem is referenced by:  mapsnf1o2  6698  mapsnf1o3  6699