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Theorem mapsncnv 6600
 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 6575 . . . . . . . . 9 (𝑥 ∈ (𝐵𝑚 {𝑋}) → 𝑥:{𝑋}⟶𝐵)
2 mapsncnv.x . . . . . . . . . 10 𝑋 ∈ V
32snid 3564 . . . . . . . . 9 𝑋 ∈ {𝑋}
4 ffvelrn 5564 . . . . . . . . 9 ((𝑥:{𝑋}⟶𝐵𝑋 ∈ {𝑋}) → (𝑥𝑋) ∈ 𝐵)
51, 3, 4sylancl 410 . . . . . . . 8 (𝑥 ∈ (𝐵𝑚 {𝑋}) → (𝑥𝑋) ∈ 𝐵)
6 eqid 2140 . . . . . . . . 9 {𝑋} = {𝑋}
7 mapsncnv.b . . . . . . . . 9 𝐵 ∈ V
86, 7, 2mapsnconst 6599 . . . . . . . 8 (𝑥 ∈ (𝐵𝑚 {𝑋}) → 𝑥 = ({𝑋} × {(𝑥𝑋)}))
95, 8jca 304 . . . . . . 7 (𝑥 ∈ (𝐵𝑚 {𝑋}) → ((𝑥𝑋) ∈ 𝐵𝑥 = ({𝑋} × {(𝑥𝑋)})))
10 eleq1 2203 . . . . . . . 8 (𝑦 = (𝑥𝑋) → (𝑦𝐵 ↔ (𝑥𝑋) ∈ 𝐵))
11 sneq 3544 . . . . . . . . . 10 (𝑦 = (𝑥𝑋) → {𝑦} = {(𝑥𝑋)})
1211xpeq2d 4574 . . . . . . . . 9 (𝑦 = (𝑥𝑋) → ({𝑋} × {𝑦}) = ({𝑋} × {(𝑥𝑋)}))
1312eqeq2d 2152 . . . . . . . 8 (𝑦 = (𝑥𝑋) → (𝑥 = ({𝑋} × {𝑦}) ↔ 𝑥 = ({𝑋} × {(𝑥𝑋)})))
1410, 13anbi12d 465 . . . . . . 7 (𝑦 = (𝑥𝑋) → ((𝑦𝐵𝑥 = ({𝑋} × {𝑦})) ↔ ((𝑥𝑋) ∈ 𝐵𝑥 = ({𝑋} × {(𝑥𝑋)}))))
159, 14syl5ibrcom 156 . . . . . 6 (𝑥 ∈ (𝐵𝑚 {𝑋}) → (𝑦 = (𝑥𝑋) → (𝑦𝐵𝑥 = ({𝑋} × {𝑦}))))
1615imp 123 . . . . 5 ((𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋)) → (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
17 fconst6g 5332 . . . . . . . . 9 (𝑦𝐵 → ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
182snex 4118 . . . . . . . . . 10 {𝑋} ∈ V
197, 18elmap 6582 . . . . . . . . 9 (({𝑋} × {𝑦}) ∈ (𝐵𝑚 {𝑋}) ↔ ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
2017, 19sylibr 133 . . . . . . . 8 (𝑦𝐵 → ({𝑋} × {𝑦}) ∈ (𝐵𝑚 {𝑋}))
21 vex 2693 . . . . . . . . . . 11 𝑦 ∈ V
2221fvconst2 5647 . . . . . . . . . 10 (𝑋 ∈ {𝑋} → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
233, 22mp1i 10 . . . . . . . . 9 (𝑦𝐵 → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
2423eqcomd 2146 . . . . . . . 8 (𝑦𝐵𝑦 = (({𝑋} × {𝑦})‘𝑋))
2520, 24jca 304 . . . . . . 7 (𝑦𝐵 → (({𝑋} × {𝑦}) ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
26 eleq1 2203 . . . . . . . 8 (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵𝑚 {𝑋}) ↔ ({𝑋} × {𝑦}) ∈ (𝐵𝑚 {𝑋})))
27 fveq1 5431 . . . . . . . . 9 (𝑥 = ({𝑋} × {𝑦}) → (𝑥𝑋) = (({𝑋} × {𝑦})‘𝑋))
2827eqeq2d 2152 . . . . . . . 8 (𝑥 = ({𝑋} × {𝑦}) → (𝑦 = (𝑥𝑋) ↔ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
2926, 28anbi12d 465 . . . . . . 7 (𝑥 = ({𝑋} × {𝑦}) → ((𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋)) ↔ (({𝑋} × {𝑦}) ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋))))
3025, 29syl5ibrcom 156 . . . . . 6 (𝑦𝐵 → (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋))))
3130imp 123 . . . . 5 ((𝑦𝐵𝑥 = ({𝑋} × {𝑦})) → (𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋)))
3216, 31impbii 125 . . . 4 ((𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
33 mapsncnv.s . . . . . . 7 𝑆 = {𝑋}
3433oveq2i 5796 . . . . . 6 (𝐵𝑚 𝑆) = (𝐵𝑚 {𝑋})
3534eleq2i 2207 . . . . 5 (𝑥 ∈ (𝐵𝑚 𝑆) ↔ 𝑥 ∈ (𝐵𝑚 {𝑋}))
3635anbi1i 454 . . . 4 ((𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑥 ∈ (𝐵𝑚 {𝑋}) ∧ 𝑦 = (𝑥𝑋)))
3733xpeq1i 4570 . . . . . 6 (𝑆 × {𝑦}) = ({𝑋} × {𝑦})
3837eqeq2i 2151 . . . . 5 (𝑥 = (𝑆 × {𝑦}) ↔ 𝑥 = ({𝑋} × {𝑦}))
3938anbi2i 453 . . . 4 ((𝑦𝐵𝑥 = (𝑆 × {𝑦})) ↔ (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
4032, 36, 393bitr4i 211 . . 3 ((𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑦𝐵𝑥 = (𝑆 × {𝑦})))
4140opabbii 4004 . 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 4000 . . . . 5 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4442, 43eqtri 2161 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4544cnveqi 4725 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))}
46 cnvopab 4951 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4745, 46eqtri 2161 . 2 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵𝑚 𝑆) ∧ 𝑦 = (𝑥𝑋))}
48 df-mpt 4000 . 2 (𝑦𝐵 ↦ (𝑆 × {𝑦})) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = (𝑆 × {𝑦}))}
4941, 47, 483eqtr4i 2171 1 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1332   ∈ wcel 1481  Vcvv 2690  {csn 3533  {copab 3997   ↦ cmpt 3998   × cxp 4548  ◡ccnv 4549  ⟶wf 5130  ‘cfv 5134  (class class class)co 5785   ↑𝑚 cmap 6553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4141  ax-un 4365  ax-setind 4462 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-v 2692  df-sbc 2915  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4225  df-xp 4556  df-rel 4557  df-cnv 4558  df-co 4559  df-dm 4560  df-rn 4561  df-res 4562  df-ima 4563  df-iota 5099  df-fun 5136  df-fn 5137  df-f 5138  df-f1 5139  df-fo 5140  df-f1o 5141  df-fv 5142  df-ov 5788  df-oprab 5789  df-mpo 5790  df-map 6555 This theorem is referenced by:  mapsnf1o2  6601  mapsnf1o3  6602
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