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Theorem mapsncnv 6722
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 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
Assertion
Ref Expression
mapsncnv 𝐹 = (𝑦𝐵 ↦ (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
StepHypRef Expression
1 elmapi 6697 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑚 {𝑋}) → 𝑥:{𝑋}⟶𝐵)
2 mapsncnv.x . . . . . . . . . 10 𝑋 ∈ V
32snid 3638 . . . . . . . . 9 𝑋 ∈ {𝑋}
4 ffvelcdm 5670 . . . . . . . . 9 ((𝑥:{𝑋}⟶𝐵𝑋 ∈ {𝑋}) → (𝑥𝑋) ∈ 𝐵)
51, 3, 4sylancl 413 . . . . . . . 8 (𝑥 ∈ (𝐵𝑚 {𝑋}) → (𝑥𝑋) ∈ 𝐵)
6 eqid 2189 . . . . . . . . 9 {𝑋} = {𝑋}
7 mapsncnv.b . . . . . . . . 9 𝐵 ∈ V
86, 7, 2mapsnconst 6721 . . . . . . . 8 (𝑥 ∈ (𝐵𝑚 {𝑋}) → 𝑥 = ({𝑋} × {(𝑥𝑋)}))
95, 8jca 306 . . . . . . 7 (𝑥 ∈ (𝐵𝑚 {𝑋}) → ((𝑥𝑋) ∈ 𝐵𝑥 = ({𝑋} × {(𝑥𝑋)})))
10 eleq1 2252 . . . . . . . 8 (𝑦 = (𝑥𝑋) → (𝑦𝐵 ↔ (𝑥𝑋) ∈ 𝐵))
11 sneq 3618 . . . . . . . . . 10 (𝑦 = (𝑥𝑋) → {𝑦} = {(𝑥𝑋)})
1211xpeq2d 4668 . . . . . . . . 9 (𝑦 = (𝑥𝑋) → ({𝑋} × {𝑦}) = ({𝑋} × {(𝑥𝑋)}))
1312eqeq2d 2201 . . . . . . . 8 (𝑦 = (𝑥𝑋) → (𝑥 = ({𝑋} × {𝑦}) ↔ 𝑥 = ({𝑋} × {(𝑥𝑋)})))
1410, 13anbi12d 473 . . . . . . 7 (𝑦 = (𝑥𝑋) → ((𝑦𝐵𝑥 = ({𝑋} × {𝑦})) ↔ ((𝑥𝑋) ∈ 𝐵𝑥 = ({𝑋} × {(𝑥𝑋)}))))
159, 14syl5ibrcom 157 . . . . . 6 (𝑥 ∈ (𝐵𝑚 {𝑋}) → (𝑦 = (𝑥𝑋) → (𝑦𝐵𝑥 = ({𝑋} × {𝑦}))))
1615imp 124 . . . . 5 ((𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋)) → (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
17 fconst6g 5433 . . . . . . . . 9 (𝑦𝐵 → ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
182snex 4203 . . . . . . . . . 10 {𝑋} ∈ V
197, 18elmap 6704 . . . . . . . . 9 (({𝑋} × {𝑦}) ∈ (𝐵𝑚 {𝑋}) ↔ ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
2017, 19sylibr 134 . . . . . . . 8 (𝑦𝐵 → ({𝑋} × {𝑦}) ∈ (𝐵𝑚 {𝑋}))
21 vex 2755 . . . . . . . . . . 11 𝑦 ∈ V
2221fvconst2 5753 . . . . . . . . . 10 (𝑋 ∈ {𝑋} → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
233, 22mp1i 10 . . . . . . . . 9 (𝑦𝐵 → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
2423eqcomd 2195 . . . . . . . 8 (𝑦𝐵𝑦 = (({𝑋} × {𝑦})‘𝑋))
2520, 24jca 306 . . . . . . 7 (𝑦𝐵 → (({𝑋} × {𝑦}) ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
26 eleq1 2252 . . . . . . . 8 (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵𝑚 {𝑋}) ↔ ({𝑋} × {𝑦}) ∈ (𝐵𝑚 {𝑋})))
27 fveq1 5533 . . . . . . . . 9 (𝑥 = ({𝑋} × {𝑦}) → (𝑥𝑋) = (({𝑋} × {𝑦})‘𝑋))
2827eqeq2d 2201 . . . . . . . 8 (𝑥 = ({𝑋} × {𝑦}) → (𝑦 = (𝑥𝑋) ↔ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
2926, 28anbi12d 473 . . . . . . 7 (𝑥 = ({𝑋} × {𝑦}) → ((𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋)) ↔ (({𝑋} × {𝑦}) ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋))))
3025, 29syl5ibrcom 157 . . . . . 6 (𝑦𝐵 → (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋))))
3130imp 124 . . . . 5 ((𝑦𝐵𝑥 = ({𝑋} × {𝑦})) → (𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋)))
3216, 31impbii 126 . . . 4 ((𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
33 mapsncnv.s . . . . . . 7 𝑆 = {𝑋}
3433oveq2i 5908 . . . . . 6 (𝐵𝑚 𝑆) = (𝐵𝑚 {𝑋})
3534eleq2i 2256 . . . . 5 (𝑥 ∈ (𝐵𝑚 𝑆) ↔ 𝑥 ∈ (𝐵𝑚 {𝑋}))
3635anbi1i 458 . . . 4 ((𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋)))
3733xpeq1i 4664 . . . . . 6 (𝑆 × {𝑦}) = ({𝑋} × {𝑦})
3837eqeq2i 2200 . . . . 5 (𝑥 = (𝑆 × {𝑦}) ↔ 𝑥 = ({𝑋} × {𝑦}))
3938anbi2i 457 . . . 4 ((𝑦𝐵𝑥 = (𝑆 × {𝑦})) ↔ (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
4032, 36, 393bitr4i 212 . . 3 ((𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑦𝐵𝑥 = (𝑆 × {𝑦})))
4140opabbii 4085 . 2 {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))} = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = (𝑆 × {𝑦}))}
42 mapsncnv.f . . . . 5 𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
43 df-mpt 4081 . . . . 5 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4442, 43eqtri 2210 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4544cnveqi 4820 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))}
46 cnvopab 5048 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4745, 46eqtri 2210 . 2 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))}
48 df-mpt 4081 . 2 (𝑦𝐵 ↦ (𝑆 × {𝑦})) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = (𝑆 × {𝑦}))}
4941, 47, 483eqtr4i 2220 1 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1364  wcel 2160  Vcvv 2752  {csn 3607  {copab 4078  cmpt 4079   × cxp 4642  ccnv 4643  wf 5231  cfv 5235  (class class class)co 5897  𝑚 cmap 6675
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-map 6677
This theorem is referenced by:  mapsnf1o2  6723  mapsnf1o3  6724
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