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Theorem mapsncnv 8918
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 8874 . . . . . . . . 9 (𝑥 ∈ (𝐵m {𝑋}) → 𝑥:{𝑋}⟶𝐵)
2 mapsncnv.x . . . . . . . . . 10 𝑋 ∈ V
32snid 4669 . . . . . . . . 9 𝑋 ∈ {𝑋}
4 ffvelcdm 7096 . . . . . . . . 9 ((𝑥:{𝑋}⟶𝐵𝑋 ∈ {𝑋}) → (𝑥𝑋) ∈ 𝐵)
51, 3, 4sylancl 584 . . . . . . . 8 (𝑥 ∈ (𝐵m {𝑋}) → (𝑥𝑋) ∈ 𝐵)
6 eqid 2728 . . . . . . . . 9 {𝑋} = {𝑋}
7 mapsncnv.b . . . . . . . . 9 𝐵 ∈ V
86, 7, 2mapsnconst 8917 . . . . . . . 8 (𝑥 ∈ (𝐵m {𝑋}) → 𝑥 = ({𝑋} × {(𝑥𝑋)}))
95, 8jca 510 . . . . . . 7 (𝑥 ∈ (𝐵m {𝑋}) → ((𝑥𝑋) ∈ 𝐵𝑥 = ({𝑋} × {(𝑥𝑋)})))
10 eleq1 2817 . . . . . . . 8 (𝑦 = (𝑥𝑋) → (𝑦𝐵 ↔ (𝑥𝑋) ∈ 𝐵))
11 sneq 4642 . . . . . . . . . 10 (𝑦 = (𝑥𝑋) → {𝑦} = {(𝑥𝑋)})
1211xpeq2d 5712 . . . . . . . . 9 (𝑦 = (𝑥𝑋) → ({𝑋} × {𝑦}) = ({𝑋} × {(𝑥𝑋)}))
1312eqeq2d 2739 . . . . . . . 8 (𝑦 = (𝑥𝑋) → (𝑥 = ({𝑋} × {𝑦}) ↔ 𝑥 = ({𝑋} × {(𝑥𝑋)})))
1410, 13anbi12d 630 . . . . . . 7 (𝑦 = (𝑥𝑋) → ((𝑦𝐵𝑥 = ({𝑋} × {𝑦})) ↔ ((𝑥𝑋) ∈ 𝐵𝑥 = ({𝑋} × {(𝑥𝑋)}))))
159, 14syl5ibrcom 246 . . . . . 6 (𝑥 ∈ (𝐵m {𝑋}) → (𝑦 = (𝑥𝑋) → (𝑦𝐵𝑥 = ({𝑋} × {𝑦}))))
1615imp 405 . . . . 5 ((𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)) → (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
17 fconst6g 6791 . . . . . . . . 9 (𝑦𝐵 → ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
18 snex 5437 . . . . . . . . . 10 {𝑋} ∈ V
197, 18elmap 8896 . . . . . . . . 9 (({𝑋} × {𝑦}) ∈ (𝐵m {𝑋}) ↔ ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
2017, 19sylibr 233 . . . . . . . 8 (𝑦𝐵 → ({𝑋} × {𝑦}) ∈ (𝐵m {𝑋}))
21 vex 3477 . . . . . . . . . . 11 𝑦 ∈ V
2221fvconst2 7222 . . . . . . . . . 10 (𝑋 ∈ {𝑋} → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
233, 22mp1i 13 . . . . . . . . 9 (𝑦𝐵 → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
2423eqcomd 2734 . . . . . . . 8 (𝑦𝐵𝑦 = (({𝑋} × {𝑦})‘𝑋))
2520, 24jca 510 . . . . . . 7 (𝑦𝐵 → (({𝑋} × {𝑦}) ∈ (𝐵m {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
26 eleq1 2817 . . . . . . . 8 (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵m {𝑋}) ↔ ({𝑋} × {𝑦}) ∈ (𝐵m {𝑋})))
27 fveq1 6901 . . . . . . . . 9 (𝑥 = ({𝑋} × {𝑦}) → (𝑥𝑋) = (({𝑋} × {𝑦})‘𝑋))
2827eqeq2d 2739 . . . . . . . 8 (𝑥 = ({𝑋} × {𝑦}) → (𝑦 = (𝑥𝑋) ↔ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
2926, 28anbi12d 630 . . . . . . 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 405 . . . . 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 7437 . . . . . 6 (𝐵m 𝑆) = (𝐵m {𝑋})
3534eleq2i 2821 . . . . 5 (𝑥 ∈ (𝐵m 𝑆) ↔ 𝑥 ∈ (𝐵m {𝑋}))
3635anbi1i 622 . . . 4 ((𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)))
3733xpeq1i 5708 . . . . . 6 (𝑆 × {𝑦}) = ({𝑋} × {𝑦})
3837eqeq2i 2741 . . . . 5 (𝑥 = (𝑆 × {𝑦}) ↔ 𝑥 = ({𝑋} × {𝑦}))
3938anbi2i 621 . . . 4 ((𝑦𝐵𝑥 = (𝑆 × {𝑦})) ↔ (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
4032, 36, 393bitr4i 302 . . 3 ((𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑦𝐵𝑥 = (𝑆 × {𝑦})))
4140opabbii 5219 . 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 5236 . . . . 5 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4442, 43eqtri 2756 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4544cnveqi 5881 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
46 cnvopab 6148 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4745, 46eqtri 2756 . 2 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
48 df-mpt 5236 . 2 (𝑦𝐵 ↦ (𝑆 × {𝑦})) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = (𝑆 × {𝑦}))}
4941, 47, 483eqtr4i 2766 1 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  Vcvv 3473  {csn 4632  {copab 5214  cmpt 5235   × cxp 5680  ccnv 5681  wf 6549  cfv 6553  (class class class)co 7426  m cmap 8851
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-map 8853
This theorem is referenced by:  mapsnf1o2  8919  mapsnf1o3  8920  coe1sfi  22139  evl1var  22262  pf1mpf  22278  pf1ind  22281  deg1val  26052
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