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Theorem mapsnf1o2 8880
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 fvex 6884 . . 3 (𝑥𝑋) ∈ V
2 mapsncnv.f . . 3 𝐹 = (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋))
31, 2fnmpti 6668 . 2 𝐹 Fn (𝐵m 𝑆)
4 mapsncnv.s . . . . 5 𝑆 = {𝑋}
5 snex 5401 . . . . 5 {𝑋} ∈ V
64, 5eqeltri 2861 . . . 4 𝑆 ∈ V
7 snex 5401 . . . 4 {𝑦} ∈ V
86, 7xpex 7740 . . 3 (𝑆 × {𝑦}) ∈ V
9 mapsncnv.b . . . 4 𝐵 ∈ V
10 mapsncnv.x . . . 4 𝑋 ∈ V
114, 9, 10, 2mapsncnv 8879 . . 3 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
128, 11fnmpti 6668 . 2 𝐹 Fn 𝐵
13 dff1o4 6819 . 2 (𝐹:(𝐵m 𝑆)–1-1-onto𝐵 ↔ (𝐹 Fn (𝐵m 𝑆) ∧ 𝐹 Fn 𝐵))
143, 12, 13mpbir2an 723 1 𝐹:(𝐵m 𝑆)–1-1-onto𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1563  wcel 2145  Vcvv 3457  {csn 4585  cmpt 5186   × cxp 5650  ccnv 5651   Fn wfn 6520  1-1-ontowf1o 6524  cfv 6525  (class class class)co 7400  m cmap 8812
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814
This theorem is referenced by:  mapsnf1o3  8881  coe1sfi  22333  coe1mul2lem2  22389  ply1coe  22419  evl1var  22457  pf1mpf  22473  pf1ind  22476  deg1ldg  26210  deg1leb  26213
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