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Theorem mapsnf1o2 6467
 Description: Explicit bijection between a set and its 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
mapsnf1o2 𝐹:(𝐵𝑚 𝑆)–1-1-onto𝐵
Distinct variable groups:   𝑥,𝐵   𝑥,𝑆
Allowed substitution hints:   𝐹(𝑥)   𝑋(𝑥)

Proof of Theorem mapsnf1o2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vex 2623 . . . 4 𝑥 ∈ V
2 mapsncnv.x . . . 4 𝑋 ∈ V
31, 2fvex 5338 . . 3 (𝑥𝑋) ∈ V
4 mapsncnv.f . . 3 𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
53, 4fnmpti 5155 . 2 𝐹 Fn (𝐵𝑚 𝑆)
6 mapsncnv.s . . . . 5 𝑆 = {𝑋}
72snex 4026 . . . . 5 {𝑋} ∈ V
86, 7eqeltri 2161 . . . 4 𝑆 ∈ V
9 vex 2623 . . . . 5 𝑦 ∈ V
109snex 4026 . . . 4 {𝑦} ∈ V
118, 10xpex 4566 . . 3 (𝑆 × {𝑦}) ∈ V
12 mapsncnv.b . . . 4 𝐵 ∈ V
136, 12, 2, 4mapsncnv 6466 . . 3 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
1411, 13fnmpti 5155 . 2 𝐹 Fn 𝐵
15 dff1o4 5274 . 2 (𝐹:(𝐵𝑚 𝑆)–1-1-onto𝐵 ↔ (𝐹 Fn (𝐵𝑚 𝑆) ∧ 𝐹 Fn 𝐵))
165, 14, 15mpbir2an 889 1 𝐹:(𝐵𝑚 𝑆)–1-1-onto𝐵
 Colors of variables: wff set class Syntax hints:   = wceq 1290   ∈ wcel 1439  Vcvv 2620  {csn 3450   ↦ cmpt 3905   × cxp 4450  ◡ccnv 4451   Fn wfn 5023  –1-1-onto→wf1o 5027  ‘cfv 5028  (class class class)co 5666   ↑𝑚 cmap 6419 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-reu 2367  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-ov 5669  df-oprab 5670  df-mpt2 5671  df-map 6421 This theorem is referenced by:  mapsnf1o3  6468
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