MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mapsnf1o2 Structured version   Visualization version   GIF version

Theorem mapsnf1o2 8447
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 6677 . . 3 (𝑥𝑋) ∈ V
2 mapsncnv.f . . 3 𝐹 = (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋))
31, 2fnmpti 6485 . 2 𝐹 Fn (𝐵m 𝑆)
4 mapsncnv.s . . . . 5 𝑆 = {𝑋}
5 snex 5323 . . . . 5 {𝑋} ∈ V
64, 5eqeltri 2909 . . . 4 𝑆 ∈ V
7 snex 5323 . . . 4 {𝑦} ∈ V
86, 7xpex 7464 . . 3 (𝑆 × {𝑦}) ∈ V
9 mapsncnv.b . . . 4 𝐵 ∈ V
10 mapsncnv.x . . . 4 𝑋 ∈ V
114, 9, 10, 2mapsncnv 8446 . . 3 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
128, 11fnmpti 6485 . 2 𝐹 Fn 𝐵
13 dff1o4 6617 . 2 (𝐹:(𝐵m 𝑆)–1-1-onto𝐵 ↔ (𝐹 Fn (𝐵m 𝑆) ∧ 𝐹 Fn 𝐵))
143, 12, 13mpbir2an 707 1 𝐹:(𝐵m 𝑆)–1-1-onto𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1528  wcel 2105  Vcvv 3495  {csn 4559  cmpt 5138   × cxp 5547  ccnv 5548   Fn wfn 6344  1-1-ontowf1o 6348  cfv 6349  (class class class)co 7145  m cmap 8396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7680  df-2nd 7681  df-map 8398
This theorem is referenced by:  mapsnf1o3  8448  coe1sfi  20311  coe1mul2lem2  20366  ply1coe  20394  evl1var  20429  pf1mpf  20445  pf1ind  20448  deg1ldg  24615  deg1leb  24618
  Copyright terms: Public domain W3C validator