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Theorem xpsff1o 16144
Description: The function appearing in xpsval 16148 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 16141 . . . . . 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 2974 . . . 4 𝑥𝐴𝑦𝐵 ({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
4 xpsff1o.f . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
54fmpt2 7183 . . . 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 7153 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
87fveq2d 6154 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
9 df-ov 6608 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
10 xp1st 7146 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (1st𝑧) ∈ 𝐴)
11 xp2nd 7147 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (2nd𝑧) ∈ 𝐵)
124xpsfval 16143 . . . . . . . . 9 (((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
1310, 11, 12syl2anc 692 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
149, 13syl5eqr 2674 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
158, 14eqtrd 2660 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
16 1st2nd2 7153 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
1716fveq2d 6154 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
18 df-ov 6608 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
19 xp1st 7146 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (1st𝑤) ∈ 𝐴)
20 xp2nd 7147 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (2nd𝑤) ∈ 𝐵)
214xpsfval 16143 . . . . . . . . 9 (((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2219, 20, 21syl2anc 692 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2318, 22syl5eqr 2674 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2417, 23eqtrd 2660 . . . . . 6 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2515, 24eqeqan12d 2642 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)})))
26 fveq1 6149 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅))
27 fvex 6160 . . . . . . . . 9 (1st𝑧) ∈ V
28 xpsc0 16136 . . . . . . . . 9 ((1st𝑧) ∈ V → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (1st𝑧))
2927, 28ax-mp 5 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (1st𝑧)
30 fvex 6160 . . . . . . . . 9 (1st𝑤) ∈ V
31 xpsc0 16136 . . . . . . . . 9 ((1st𝑤) ∈ V → (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅) = (1st𝑤))
3230, 31ax-mp 5 . . . . . . . 8 (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅) = (1st𝑤)
3326, 29, 323eqtr3g 2683 . . . . . . 7 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (1st𝑧) = (1st𝑤))
34 fveq1 6149 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1𝑜) = (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1𝑜))
35 fvex 6160 . . . . . . . . 9 (2nd𝑧) ∈ V
36 xpsc1 16137 . . . . . . . . 9 ((2nd𝑧) ∈ V → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1𝑜) = (2nd𝑧))
3735, 36ax-mp 5 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1𝑜) = (2nd𝑧)
38 fvex 6160 . . . . . . . . 9 (2nd𝑤) ∈ V
39 xpsc1 16137 . . . . . . . . 9 ((2nd𝑤) ∈ V → (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1𝑜) = (2nd𝑤))
4038, 39ax-mp 5 . . . . . . . 8 (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1𝑜) = (2nd𝑤)
4134, 37, 403eqtr3g 2683 . . . . . . 7 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (2nd𝑧) = (2nd𝑤))
4233, 41opeq12d 4383 . . . . . 6 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩)
437, 16eqeqan12d 2642 . . . . . 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 2974 . . 3 𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
47 dff13 6467 . . 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 954 . 2 𝐹:(𝐴 × 𝐵)–1-1X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
49 xpsfrnel 16139 . . . . . 6 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2𝑜 ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1𝑜) ∈ 𝐵))
5049simp2bi 1075 . . . . 5 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
5149simp3bi 1076 . . . . 5 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1𝑜) ∈ 𝐵)
524xpsfval 16143 . . . . . . 7 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1𝑜) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1𝑜)) = ({(𝑧‘∅)} +𝑐 {(𝑧‘1𝑜)}))
5350, 51, 52syl2anc 692 . . . . . 6 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1𝑜)) = ({(𝑧‘∅)} +𝑐 {(𝑧‘1𝑜)}))
54 ixpfn 7859 . . . . . . 7 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2𝑜)
55 xpsfeq 16140 . . . . . . 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 2661 . . . . 5 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1𝑜)))
58 rspceov 6646 . . . . 5 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1𝑜) ∈ 𝐵𝑧 = ((𝑧‘∅)𝐹(𝑧‘1𝑜))) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
5950, 51, 57, 58syl3anc 1323 . . . 4 (𝑧X𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
6059rgen 2922 . . 3 𝑧X 𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏)
61 foov 6762 . . 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 954 . 2 𝐹:(𝐴 × 𝐵)–ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
63 df-f1o 5857 . 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 954 1 𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2𝑜 if(𝑘 = ∅, 𝐴, 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  wrex 2913  Vcvv 3191  c0 3896  ifcif 4063  {csn 4153  cop 4159   × cxp 5077  ccnv 5078   Fn wfn 5845  wf 5846  1-1wf1 5847  ontowfo 5848  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  cmpt2 6607  1st c1st 7114  2nd c2nd 7115  1𝑜c1o 7499  2𝑜c2o 7500  Xcixp 7853   +𝑐 ccda 8934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-om 7014  df-1st 7116  df-2nd 7117  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-1o 7506  df-2o 7507  df-oadd 7510  df-er 7688  df-ixp 7854  df-en 7901  df-dom 7902  df-sdom 7903  df-fin 7904  df-cda 8935
This theorem is referenced by:  xpsfrn  16145  xpsff1o2  16147
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