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Theorem mapsnf1o3 8836
Description: Explicit bijection in the reverse of mapsnf1o2 8835. (Contributed by Stefan O'Rear, 24-Mar-2015.)
Hypotheses
Ref Expression
mapsncnv.s 𝑆 = {𝑋}
mapsncnv.b 𝐵 ∈ V
mapsncnv.x 𝑋 ∈ V
mapsnf1o3.f 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
Assertion
Ref Expression
mapsnf1o3 𝐹:𝐵1-1-onto→(𝐵m 𝑆)
Distinct variable groups:   𝑦,𝐵   𝑦,𝑆   𝑦,𝑋
Allowed substitution hint:   𝐹(𝑦)

Proof of Theorem mapsnf1o3
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 mapsncnv.s . . . 4 𝑆 = {𝑋}
2 mapsncnv.b . . . 4 𝐵 ∈ V
3 mapsncnv.x . . . 4 𝑋 ∈ V
4 eqid 2733 . . . 4 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)) = (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋))
51, 2, 3, 4mapsnf1o2 8835 . . 3 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)):(𝐵m 𝑆)–1-1-onto𝐵
6 f1ocnv 6797 . . 3 ((𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)):(𝐵m 𝑆)–1-1-onto𝐵(𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)):𝐵1-1-onto→(𝐵m 𝑆))
75, 6ax-mp 5 . 2 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)):𝐵1-1-onto→(𝐵m 𝑆)
8 mapsnf1o3.f . . . 4 𝐹 = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
91, 2, 3, 4mapsncnv 8834 . . . 4 (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)) = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
108, 9eqtr4i 2764 . . 3 𝐹 = (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋))
11 f1oeq1 6773 . . 3 (𝐹 = (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)) → (𝐹:𝐵1-1-onto→(𝐵m 𝑆) ↔ (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)):𝐵1-1-onto→(𝐵m 𝑆)))
1210, 11ax-mp 5 . 2 (𝐹:𝐵1-1-onto→(𝐵m 𝑆) ↔ (𝑥 ∈ (𝐵m 𝑆) ↦ (𝑥𝑋)):𝐵1-1-onto→(𝐵m 𝑆))
137, 12mpbir 230 1 𝐹:𝐵1-1-onto→(𝐵m 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  Vcvv 3444  {csn 4587  cmpt 5189   × cxp 5632  ccnv 5633  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  m cmap 8768
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770
This theorem is referenced by:  coe1f2  21596  coe1add  21651  evls1rhmlem  21703  evl1sca  21716  pf1ind  21737  ismrer1  36343
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