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Theorem mapsnf1o2 8884
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 6901 . . 3 (𝑥𝑋) ∈ V
2 mapsncnv.f . . 3 𝐹 = (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋))
31, 2fnmpti 6690 . 2 𝐹 Fn (𝐵m 𝑆)
4 mapsncnv.s . . . . 5 𝑆 = {𝑋}
5 snex 5430 . . . . 5 {𝑋} ∈ V
64, 5eqeltri 2830 . . . 4 𝑆 ∈ V
7 snex 5430 . . . 4 {𝑦} ∈ V
86, 7xpex 7735 . . 3 (𝑆 × {𝑦}) ∈ V
9 mapsncnv.b . . . 4 𝐵 ∈ V
10 mapsncnv.x . . . 4 𝑋 ∈ V
114, 9, 10, 2mapsncnv 8883 . . 3 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
128, 11fnmpti 6690 . 2 𝐹 Fn 𝐵
13 dff1o4 6838 . 2 (𝐹:(𝐵m 𝑆)–1-1-onto𝐵 ↔ (𝐹 Fn (𝐵m 𝑆) ∧ 𝐹 Fn 𝐵))
143, 12, 13mpbir2an 710 1 𝐹:(𝐵m 𝑆)–1-1-onto𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  Vcvv 3475  {csn 4627  cmpt 5230   × cxp 5673  ccnv 5674   Fn wfn 6535  1-1-ontowf1o 6539  cfv 6540  (class class class)co 7404  m cmap 8816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7970  df-2nd 7971  df-map 8818
This theorem is referenced by:  mapsnf1o3  8885  coe1sfi  21719  coe1mul2lem2  21772  ply1coe  21802  evl1var  21837  pf1mpf  21853  pf1ind  21856  deg1ldg  25592  deg1leb  25595
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