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Theorem xpsff1o 16275
Description: The function appearing in xpsval 16279 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 16272 . . . . . 6 (({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥𝐴𝑦𝐵))
21biimpri 218 . . . . 5 ((𝑥𝐴𝑦𝐵) → ({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵))
32rgen2 3004 . . . 4 𝑥𝐴𝑦𝐵 ({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
4 xpsff1o.f . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
54fmpt2 7282 . . . 4 (∀𝑥𝐴𝑦𝐵 ({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵))
63, 5mpbi 220 . . 3 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
7 1st2nd2 7249 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
87fveq2d 6233 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
9 df-ov 6693 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
10 xp1st 7242 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (1st𝑧) ∈ 𝐴)
11 xp2nd 7243 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (2nd𝑧) ∈ 𝐵)
124xpsfval 16274 . . . . . . . . 9 (((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
1310, 11, 12syl2anc 694 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
149, 13syl5eqr 2699 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
158, 14eqtrd 2685 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
16 1st2nd2 7249 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
1716fveq2d 6233 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
18 df-ov 6693 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
19 xp1st 7242 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (1st𝑤) ∈ 𝐴)
20 xp2nd 7243 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (2nd𝑤) ∈ 𝐵)
214xpsfval 16274 . . . . . . . . 9 (((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2219, 20, 21syl2anc 694 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2318, 22syl5eqr 2699 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2417, 23eqtrd 2685 . . . . . 6 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2515, 24eqeqan12d 2667 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)})))
26 fveq1 6228 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅))
27 fvex 6239 . . . . . . . . 9 (1st𝑧) ∈ V
28 xpsc0 16267 . . . . . . . . 9 ((1st𝑧) ∈ V → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (1st𝑧))
2927, 28ax-mp 5 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (1st𝑧)
30 fvex 6239 . . . . . . . . 9 (1st𝑤) ∈ V
31 xpsc0 16267 . . . . . . . . 9 ((1st𝑤) ∈ V → (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅) = (1st𝑤))
3230, 31ax-mp 5 . . . . . . . 8 (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅) = (1st𝑤)
3326, 29, 323eqtr3g 2708 . . . . . . 7 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (1st𝑧) = (1st𝑤))
34 fveq1 6228 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1𝑜) = (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1𝑜))
35 fvex 6239 . . . . . . . . 9 (2nd𝑧) ∈ V
36 xpsc1 16268 . . . . . . . . 9 ((2nd𝑧) ∈ V → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1𝑜) = (2nd𝑧))
3735, 36ax-mp 5 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1𝑜) = (2nd𝑧)
38 fvex 6239 . . . . . . . . 9 (2nd𝑤) ∈ V
39 xpsc1 16268 . . . . . . . . 9 ((2nd𝑤) ∈ V → (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1𝑜) = (2nd𝑤))
4038, 39ax-mp 5 . . . . . . . 8 (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1𝑜) = (2nd𝑤)
4134, 37, 403eqtr3g 2708 . . . . . . 7 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (2nd𝑧) = (2nd𝑤))
4233, 41opeq12d 4441 . . . . . 6 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩)
437, 16eqeqan12d 2667 . . . . . 6 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
4442, 43syl5ibr 236 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → 𝑧 = 𝑤))
4525, 44sylbid 230 . . . 4 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
4645rgen2 3004 . . 3 𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
47 dff13 6552 . . 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 975 . 2 𝐹:(𝐴 × 𝐵)–1-1X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
49 xpsfrnel 16270 . . . . . 6 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2𝑜 ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1𝑜) ∈ 𝐵))
5049simp2bi 1097 . . . . 5 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
5149simp3bi 1098 . . . . 5 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1𝑜) ∈ 𝐵)
524xpsfval 16274 . . . . . . 7 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1𝑜) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1𝑜)) = ({(𝑧‘∅)} +𝑐 {(𝑧‘1𝑜)}))
5350, 51, 52syl2anc 694 . . . . . 6 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1𝑜)) = ({(𝑧‘∅)} +𝑐 {(𝑧‘1𝑜)}))
54 ixpfn 7956 . . . . . . 7 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2𝑜)
55 xpsfeq 16271 . . . . . . 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 2686 . . . . 5 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1𝑜)))
58 rspceov 6732 . . . . 5 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1𝑜) ∈ 𝐵𝑧 = ((𝑧‘∅)𝐹(𝑧‘1𝑜))) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
5950, 51, 57, 58syl3anc 1366 . . . 4 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
6059rgen 2951 . . 3 𝑧X 𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏)
61 foov 6850 . . 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 975 . 2 𝐹:(𝐴 × 𝐵)–ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
63 df-f1o 5933 . 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 975 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 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  c0 3948  ifcif 4119  {csn 4210  cop 4216   × cxp 5141  ccnv 5142   Fn wfn 5921  wf 5922  1-1wf1 5923  ontowfo 5924  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  1𝑜c1o 7598  2𝑜c2o 7599  Xcixp 7950   +𝑐 ccda 9027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-cda 9028
This theorem is referenced by:  xpsfrn  16276  xpsff1o2  16278
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