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Theorem mapsnf1o3 7792
 Description: Explicit bijection in the reverse of mapsnf1o2 7791. (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 2610 . . . 4 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)) = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
51, 2, 3, 4mapsnf1o2 7791 . . 3 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)):(𝐵𝑚 𝑆)–1-1-onto𝐵
6 f1ocnv 6062 . . 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 7790 . . . 4 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)) = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
108, 9eqtr4i 2635 . . 3 𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
11 f1oeq1 6040 . . 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 220 1 𝐹:𝐵1-1-onto→(𝐵𝑚 𝑆)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746 This theorem is referenced by:  coe1f2  19400  coe1add  19455  evls1rhmlem  19507  evl1sca  19519  pf1ind  19540  ismrer1  32807
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