Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpsff1o Structured version   Visualization version   GIF version

Theorem xpsff1o 16051
 Description: The function appearing in xpsval 16055 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2𝑜 = {∅, 1𝑜}. (Contributed by Mario Carneiro, 15-Aug-2015.)
Hypothesis
Ref Expression
xpsff1o.f 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
Assertion
Ref Expression
xpsff1o 𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
Distinct variable groups:   𝑥,𝑘,𝑦,𝐴   𝐵,𝑘,𝑥,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑘)

Proof of Theorem xpsff1o
Dummy variables 𝑎 𝑏 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpsfrnel2 16048 . . . . . 6 (({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥𝐴𝑦𝐵))
21biimpri 217 . . . . 5 ((𝑥𝐴𝑦𝐵) → ({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵))
32rgen2 2958 . . . 4 𝑥𝐴𝑦𝐵 ({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
4 xpsff1o.f . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
54fmpt2 7126 . . . 4 (∀𝑥𝐴𝑦𝐵 ({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵))
63, 5mpbi 219 . . 3 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
7 1st2nd2 7096 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
87fveq2d 6107 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
9 df-ov 6552 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
10 xp1st 7089 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (1st𝑧) ∈ 𝐴)
11 xp2nd 7090 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (2nd𝑧) ∈ 𝐵)
124xpsfval 16050 . . . . . . . . 9 (((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
1310, 11, 12syl2anc 691 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
149, 13syl5eqr 2658 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
158, 14eqtrd 2644 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
16 1st2nd2 7096 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
1716fveq2d 6107 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
18 df-ov 6552 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
19 xp1st 7089 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (1st𝑤) ∈ 𝐴)
20 xp2nd 7090 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (2nd𝑤) ∈ 𝐵)
214xpsfval 16050 . . . . . . . . 9 (((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2219, 20, 21syl2anc 691 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2318, 22syl5eqr 2658 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2417, 23eqtrd 2644 . . . . . 6 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2515, 24eqeqan12d 2626 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)})))
26 fveq1 6102 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅))
27 fvex 6113 . . . . . . . . 9 (1st𝑧) ∈ V
28 xpsc0 16043 . . . . . . . . 9 ((1st𝑧) ∈ V → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (1st𝑧))
2927, 28ax-mp 5 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (1st𝑧)
30 fvex 6113 . . . . . . . . 9 (1st𝑤) ∈ V
31 xpsc0 16043 . . . . . . . . 9 ((1st𝑤) ∈ V → (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅) = (1st𝑤))
3230, 31ax-mp 5 . . . . . . . 8 (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅) = (1st𝑤)
3326, 29, 323eqtr3g 2667 . . . . . . 7 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (1st𝑧) = (1st𝑤))
34 fveq1 6102 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1𝑜) = (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1𝑜))
35 fvex 6113 . . . . . . . . 9 (2nd𝑧) ∈ V
36 xpsc1 16044 . . . . . . . . 9 ((2nd𝑧) ∈ V → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1𝑜) = (2nd𝑧))
3735, 36ax-mp 5 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1𝑜) = (2nd𝑧)
38 fvex 6113 . . . . . . . . 9 (2nd𝑤) ∈ V
39 xpsc1 16044 . . . . . . . . 9 ((2nd𝑤) ∈ V → (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1𝑜) = (2nd𝑤))
4038, 39ax-mp 5 . . . . . . . 8 (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1𝑜) = (2nd𝑤)
4134, 37, 403eqtr3g 2667 . . . . . . 7 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (2nd𝑧) = (2nd𝑤))
4233, 41opeq12d 4348 . . . . . 6 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩)
437, 16eqeqan12d 2626 . . . . . 6 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
4442, 43syl5ibr 235 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → 𝑧 = 𝑤))
4525, 44sylbid 229 . . . 4 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
4645rgen2 2958 . . 3 𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
47 dff13 6416 . . 3 (𝐹:(𝐴 × 𝐵)–1-1X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ∧ ∀𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)))
486, 46, 47mpbir2an 957 . 2 𝐹:(𝐴 × 𝐵)–1-1X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
49 xpsfrnel 16046 . . . . . 6 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2𝑜 ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1𝑜) ∈ 𝐵))
5049simp2bi 1070 . . . . 5 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
5149simp3bi 1071 . . . . 5 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1𝑜) ∈ 𝐵)
524xpsfval 16050 . . . . . . 7 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1𝑜) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1𝑜)) = ({(𝑧‘∅)} +𝑐 {(𝑧‘1𝑜)}))
5350, 51, 52syl2anc 691 . . . . . 6 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1𝑜)) = ({(𝑧‘∅)} +𝑐 {(𝑧‘1𝑜)}))
54 ixpfn 7800 . . . . . . 7 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2𝑜)
55 xpsfeq 16047 . . . . . . 7 (𝑧 Fn 2𝑜({(𝑧‘∅)} +𝑐 {(𝑧‘1𝑜)}) = 𝑧)
5654, 55syl 17 . . . . . 6 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → ({(𝑧‘∅)} +𝑐 {(𝑧‘1𝑜)}) = 𝑧)
5753, 56eqtr2d 2645 . . . . 5 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1𝑜)))
58 rspceov 6590 . . . . 5 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1𝑜) ∈ 𝐵𝑧 = ((𝑧‘∅)𝐹(𝑧‘1𝑜))) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
5950, 51, 57, 58syl3anc 1318 . . . 4 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
6059rgen 2906 . . 3 𝑧X 𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏)
61 foov 6706 . . 3 (𝐹:(𝐴 × 𝐵)–ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ∧ ∀𝑧X 𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏)))
626, 60, 61mpbir2an 957 . 2 𝐹:(𝐴 × 𝐵)–ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
63 df-f1o 5811 . 2 (𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝐹:(𝐴 × 𝐵)–1-1X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ∧ 𝐹:(𝐴 × 𝐵)–ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)))
6448, 62, 63mpbir2an 957 1 𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173  ∅c0 3874  ifcif 4036  {csn 4125  ⟨cop 4131   × cxp 5036  ◡ccnv 5037   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –onto→wfo 5802  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  1𝑜c1o 7440  2𝑜c2o 7441  Xcixp 7794   +𝑐 ccda 8872 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-3or 1032  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-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  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-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  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-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-cda 8873 This theorem is referenced by:  xpsfrn  16052  xpsff1o2  16054
 Copyright terms: Public domain W3C validator