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Theorem mapsncnv 8449
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 8420 . . . . . . . . 9 (𝑥 ∈ (𝐵m {𝑋}) → 𝑥:{𝑋}⟶𝐵)
2 mapsncnv.x . . . . . . . . . 10 𝑋 ∈ V
32snid 4593 . . . . . . . . 9 𝑋 ∈ {𝑋}
4 ffvelrn 6842 . . . . . . . . 9 ((𝑥:{𝑋}⟶𝐵𝑋 ∈ {𝑋}) → (𝑥𝑋) ∈ 𝐵)
51, 3, 4sylancl 588 . . . . . . . 8 (𝑥 ∈ (𝐵m {𝑋}) → (𝑥𝑋) ∈ 𝐵)
6 eqid 2819 . . . . . . . . 9 {𝑋} = {𝑋}
7 mapsncnv.b . . . . . . . . 9 𝐵 ∈ V
86, 7, 2mapsnconst 8448 . . . . . . . 8 (𝑥 ∈ (𝐵m {𝑋}) → 𝑥 = ({𝑋} × {(𝑥𝑋)}))
95, 8jca 514 . . . . . . 7 (𝑥 ∈ (𝐵m {𝑋}) → ((𝑥𝑋) ∈ 𝐵𝑥 = ({𝑋} × {(𝑥𝑋)})))
10 eleq1 2898 . . . . . . . 8 (𝑦 = (𝑥𝑋) → (𝑦𝐵 ↔ (𝑥𝑋) ∈ 𝐵))
11 sneq 4569 . . . . . . . . . 10 (𝑦 = (𝑥𝑋) → {𝑦} = {(𝑥𝑋)})
1211xpeq2d 5578 . . . . . . . . 9 (𝑦 = (𝑥𝑋) → ({𝑋} × {𝑦}) = ({𝑋} × {(𝑥𝑋)}))
1312eqeq2d 2830 . . . . . . . 8 (𝑦 = (𝑥𝑋) → (𝑥 = ({𝑋} × {𝑦}) ↔ 𝑥 = ({𝑋} × {(𝑥𝑋)})))
1410, 13anbi12d 632 . . . . . . 7 (𝑦 = (𝑥𝑋) → ((𝑦𝐵𝑥 = ({𝑋} × {𝑦})) ↔ ((𝑥𝑋) ∈ 𝐵𝑥 = ({𝑋} × {(𝑥𝑋)}))))
159, 14syl5ibrcom 249 . . . . . 6 (𝑥 ∈ (𝐵m {𝑋}) → (𝑦 = (𝑥𝑋) → (𝑦𝐵𝑥 = ({𝑋} × {𝑦}))))
1615imp 409 . . . . 5 ((𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)) → (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
17 fconst6g 6561 . . . . . . . . 9 (𝑦𝐵 → ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
18 snex 5322 . . . . . . . . . 10 {𝑋} ∈ V
197, 18elmap 8427 . . . . . . . . 9 (({𝑋} × {𝑦}) ∈ (𝐵m {𝑋}) ↔ ({𝑋} × {𝑦}):{𝑋}⟶𝐵)
2017, 19sylibr 236 . . . . . . . 8 (𝑦𝐵 → ({𝑋} × {𝑦}) ∈ (𝐵m {𝑋}))
21 vex 3496 . . . . . . . . . . 11 𝑦 ∈ V
2221fvconst2 6959 . . . . . . . . . 10 (𝑋 ∈ {𝑋} → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
233, 22mp1i 13 . . . . . . . . 9 (𝑦𝐵 → (({𝑋} × {𝑦})‘𝑋) = 𝑦)
2423eqcomd 2825 . . . . . . . 8 (𝑦𝐵𝑦 = (({𝑋} × {𝑦})‘𝑋))
2520, 24jca 514 . . . . . . 7 (𝑦𝐵 → (({𝑋} × {𝑦}) ∈ (𝐵m {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
26 eleq1 2898 . . . . . . . 8 (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵m {𝑋}) ↔ ({𝑋} × {𝑦}) ∈ (𝐵m {𝑋})))
27 fveq1 6662 . . . . . . . . 9 (𝑥 = ({𝑋} × {𝑦}) → (𝑥𝑋) = (({𝑋} × {𝑦})‘𝑋))
2827eqeq2d 2830 . . . . . . . 8 (𝑥 = ({𝑋} × {𝑦}) → (𝑦 = (𝑥𝑋) ↔ 𝑦 = (({𝑋} × {𝑦})‘𝑋)))
2926, 28anbi12d 632 . . . . . . 7 (𝑥 = ({𝑋} × {𝑦}) → ((𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)) ↔ (({𝑋} × {𝑦}) ∈ (𝐵m {𝑋}) ∧ 𝑦 = (({𝑋} × {𝑦})‘𝑋))))
3025, 29syl5ibrcom 249 . . . . . 6 (𝑦𝐵 → (𝑥 = ({𝑋} × {𝑦}) → (𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋))))
3130imp 409 . . . . 5 ((𝑦𝐵𝑥 = ({𝑋} × {𝑦})) → (𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)))
3216, 31impbii 211 . . . 4 ((𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
33 mapsncnv.s . . . . . . 7 𝑆 = {𝑋}
3433oveq2i 7159 . . . . . 6 (𝐵m 𝑆) = (𝐵m {𝑋})
3534eleq2i 2902 . . . . 5 (𝑥 ∈ (𝐵m 𝑆) ↔ 𝑥 ∈ (𝐵m {𝑋}))
3635anbi1i 625 . . . 4 ((𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑥 ∈ (𝐵m {𝑋}) ∧ 𝑦 = (𝑥𝑋)))
3733xpeq1i 5574 . . . . . 6 (𝑆 × {𝑦}) = ({𝑋} × {𝑦})
3837eqeq2i 2832 . . . . 5 (𝑥 = (𝑆 × {𝑦}) ↔ 𝑥 = ({𝑋} × {𝑦}))
3938anbi2i 624 . . . 4 ((𝑦𝐵𝑥 = (𝑆 × {𝑦})) ↔ (𝑦𝐵𝑥 = ({𝑋} × {𝑦})))
4032, 36, 393bitr4i 305 . . 3 ((𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋)) ↔ (𝑦𝐵𝑥 = (𝑆 × {𝑦})))
4140opabbii 5124 . 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 5138 . . . . 5 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4442, 43eqtri 2842 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4544cnveqi 5738 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
46 cnvopab 5990 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
4745, 46eqtri 2842 . 2 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ (𝐵m 𝑆) ∧ 𝑦 = (𝑥𝑋))}
48 df-mpt 5138 . 2 (𝑦𝐵 ↦ (𝑆 × {𝑦})) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐵𝑥 = (𝑆 × {𝑦}))}
4941, 47, 483eqtr4i 2852 1 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1530  wcel 2107  Vcvv 3493  {csn 4559  {copab 5119  cmpt 5137   × cxp 5546  ccnv 5547  wf 6344  cfv 6348  (class class class)co 7148  m cmap 8398
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-map 8400
This theorem is referenced by:  mapsnf1o2  8450  mapsnf1o3  8451  coe1sfi  20373  evl1var  20491  pf1mpf  20507  pf1ind  20510  deg1val  24682
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