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Theorem mapsnf1o3 8249
Description: Explicit bijection in the reverse of mapsnf1o2 8248. (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 2772 . . . 4 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)) = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
51, 2, 3, 4mapsnf1o2 8248 . . 3 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)):(𝐵𝑚 𝑆)–1-1-onto𝐵
6 f1ocnv 6450 . . 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 8247 . . . 4 (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋)) = (𝑦𝐵 ↦ (𝑆 × {𝑦}))
108, 9eqtr4i 2799 . . 3 𝐹 = (𝑥 ∈ (𝐵𝑚 𝑆) ↦ (𝑥𝑋))
11 f1oeq1 6427 . . 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 223 1 𝐹:𝐵1-1-onto→(𝐵𝑚 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1507  wcel 2048  Vcvv 3409  {csn 4435  cmpt 5002   × cxp 5398  ccnv 5399  1-1-ontowf1o 6181  cfv 6182  (class class class)co 6970  𝑚 cmap 8198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-map 8200
This theorem is referenced by:  coe1f2  20070  coe1add  20125  evls1rhmlem  20177  evl1sca  20189  pf1ind  20210  ismrer1  34506
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