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Theorem mapsncnv 8831
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 8787 . . . . . . . . 9 (𝑥 ∈ (𝐵m {𝑋}) → 𝑥:{𝑋}⟶𝐵)
2 mapsncnv.x . . . . . . . . . 10 𝑋 ∈ V
32snid 4622 . . . . . . . . 9 𝑋 ∈ {𝑋}
4 ffvelcdm 7032 . . . . . . . . 9 ((𝑥:{𝑋}⟶𝐵𝑋 ∈ {𝑋}) → (𝑥𝑋) ∈ 𝐵)
51, 3, 4sylancl 586 . . . . . . . 8 (𝑥 ∈ (𝐵m {𝑋}) → (𝑥𝑋) ∈ 𝐵)
6 eqid 2736 . . . . . . . . 9 {𝑋} = {𝑋}
7 mapsncnv.b . . . . . . . . 9 𝐵 ∈ V
86, 7, 2mapsnconst 8830 . . . . . . . 8 (𝑥 ∈ (𝐵m {𝑋}) → 𝑥 = ({𝑋} × {(𝑥𝑋)}))
95, 8jca 512 . . . . . . 7 (𝑥 ∈ (𝐵m {𝑋}) → ((𝑥𝑋) ∈ 𝐵𝑥 = ({𝑋} × {(𝑥𝑋)})))
10 eleq1 2825 . . . . . . . 8 (𝑦 = (𝑥𝑋) → (𝑦𝐵 ↔ (𝑥𝑋) ∈ 𝐵))
11 sneq 4596 . . . . . . . . . 10 (𝑦 = (𝑥𝑋) → {𝑦} = {(𝑥𝑋)})
1211xpeq2d 5663 . . . . . . . . 9 (𝑦 = (𝑥𝑋) → ({𝑋} × {𝑦}) = ({𝑋} × {(𝑥𝑋)}))
1312eqeq2d 2747 . . . . . . . 8 (𝑦 = (𝑥𝑋) → (𝑥 = ({𝑋} × {𝑦}) ↔ 𝑥 = ({𝑋} × {(𝑥𝑋)})))
1410, 13anbi12d 631 . . . . . . 7 (𝑦 = (𝑥𝑋) → ((𝑦𝐵𝑥 = ({𝑋} × {𝑦})) ↔ ((𝑥𝑋) ∈ 𝐵𝑥 = ({𝑋} × {(𝑥𝑋)}))))
159, 14syl5ibrcom 246 . . . . . 6 (𝑥 ∈ (𝐵m {𝑋}) → (𝑦 = (𝑥𝑋) → (𝑦𝐵𝑥 = ({𝑋} × {𝑦}))))
1615imp 407 . . . . 5 ((𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)) → (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
17 fconst6g 6731 . . . . . . . . 9 (𝑦𝐵 → ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
18 snex 5388 . . . . . . . . . 10 {𝑋} ∈ V
197, 18elmap 8809 . . . . . . . . 9 (({𝑋} × {𝑦}) ∈ (𝐵m {𝑋}) ↔ ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
2017, 19sylibr 233 . . . . . . . 8 (𝑦𝐵 → ({𝑋} × {𝑦}) ∈ (𝐵m {𝑋}))
21 vex 3449 . . . . . . . . . . 11 𝑦 ∈ V
2221fvconst2 7153 . . . . . . . . . 10 (𝑋 ∈ {𝑋} → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
233, 22mp1i 13 . . . . . . . . 9 (𝑦𝐵 → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
2423eqcomd 2742 . . . . . . . 8 (𝑦𝐵𝑦 = (({𝑋} × {𝑦})‘𝑋))
2520, 24jca 512 . . . . . . 7 (𝑦𝐵 → (({𝑋} × {𝑦}) ∈ (𝐵m {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
26 eleq1 2825 . . . . . . . 8 (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵m {𝑋}) ↔ ({𝑋} × {𝑦}) ∈ (𝐵m {𝑋})))
27 fveq1 6841 . . . . . . . . 9 (𝑥 = ({𝑋} × {𝑦}) → (𝑥𝑋) = (({𝑋} × {𝑦})‘𝑋))
2827eqeq2d 2747 . . . . . . . 8 (𝑥 = ({𝑋} × {𝑦}) → (𝑦 = (𝑥𝑋) ↔ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
2926, 28anbi12d 631 . . . . . . 7 (𝑥 = ({𝑋} × {𝑦}) → ((𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)) ↔ (({𝑋} × {𝑦}) ∈ (𝐵m {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋))))
3025, 29syl5ibrcom 246 . . . . . 6 (𝑦𝐵 → (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋))))
3130imp 407 . . . . 5 ((𝑦𝐵𝑥 = ({𝑋} × {𝑦})) → (𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)))
3216, 31impbii 208 . . . 4 ((𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
33 mapsncnv.s . . . . . . 7 𝑆 = {𝑋}
3433oveq2i 7368 . . . . . 6 (𝐵m 𝑆) = (𝐵m {𝑋})
3534eleq2i 2829 . . . . 5 (𝑥 ∈ (𝐵m 𝑆) ↔ 𝑥 ∈ (𝐵m {𝑋}))
3635anbi1i 624 . . . 4 ((𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)))
3733xpeq1i 5659 . . . . . 6 (𝑆 × {𝑦}) = ({𝑋} × {𝑦})
3837eqeq2i 2749 . . . . 5 (𝑥 = (𝑆 × {𝑦}) ↔ 𝑥 = ({𝑋} × {𝑦}))
3938anbi2i 623 . . . 4 ((𝑦𝐵𝑥 = (𝑆 × {𝑦})) ↔ (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
4032, 36, 393bitr4i 302 . . 3 ((𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑦𝐵𝑥 = (𝑆 × {𝑦})))
4140opabbii 5172 . 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 5189 . . . . 5 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4442, 43eqtri 2764 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4544cnveqi 5830 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
46 cnvopab 6091 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4745, 46eqtri 2764 . 2 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
48 df-mpt 5189 . 2 (𝑦𝐵 ↦ (𝑆 × {𝑦})) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = (𝑆 × {𝑦}))}
4941, 47, 483eqtr4i 2774 1 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wcel 2106  Vcvv 3445  {csn 4586  {copab 5167  cmpt 5188   × cxp 5631  ccnv 5632  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767
This theorem is referenced by:  mapsnf1o2  8832  mapsnf1o3  8833  coe1sfi  21584  evl1var  21702  pf1mpf  21718  pf1ind  21721  deg1val  25461
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