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Theorem mapsnf1o2 6597
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 2692 . . . 4 𝑥 ∈ V
2 mapsncnv.x . . . 4 𝑋 ∈ V
31, 2fvex 5448 . . 3 (𝑥𝑋) ∈ V
4 mapsncnv.f . . 3 𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
53, 4fnmpti 5258 . 2 𝐹 Fn (𝐵𝑚 𝑆)
6 mapsncnv.s . . . . 5 𝑆 = {𝑋}
72snex 4116 . . . . 5 {𝑋} ∈ V
86, 7eqeltri 2213 . . . 4 𝑆 ∈ V
9 vex 2692 . . . . 5 𝑦 ∈ V
109snex 4116 . . . 4 {𝑦} ∈ V
118, 10xpex 4661 . . 3 (𝑆 × {𝑦}) ∈ V
12 mapsncnv.b . . . 4 𝐵 ∈ V
136, 12, 2, 4mapsncnv 6596 . . 3 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
1411, 13fnmpti 5258 . 2 𝐹 Fn 𝐵
15 dff1o4 5382 . 2 (𝐹:(𝐵𝑚 𝑆)–1-1-onto𝐵 ↔ (𝐹 Fn (𝐵𝑚 𝑆) ∧ 𝐹 Fn 𝐵))
165, 14, 15mpbir2an 927 1 𝐹:(𝐵𝑚 𝑆)–1-1-onto𝐵
Colors of variables: wff set class
Syntax hints:   = wceq 1332  wcel 1481  Vcvv 2689  {csn 3531  cmpt 3996   × cxp 4544  ccnv 4545   Fn wfn 5125  1-1-ontowf1o 5129  cfv 5130  (class class class)co 5781  𝑚 cmap 6549
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 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459
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 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-map 6551
This theorem is referenced by:  mapsnf1o3  6598
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