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Theorem xpsff1o 16834
 Description: The function appearing in xpsval 16837 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⟩, ⟨1o, 𝑦⟩})
Assertion
Ref Expression
xpsff1o 𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
Distinct variable groups:   𝐴,𝑘,𝑥,𝑦   𝐵,𝑘,𝑥,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦,𝑘)

Proof of Theorem xpsff1o
Dummy variables 𝑎 𝑏 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpsfrnel2 16831 . . . . . 6 ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥𝐴𝑦𝐵))
21biimpri 230 . . . . 5 ((𝑥𝐴𝑦𝐵) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵))
32rgen2 3203 . . . 4 𝑥𝐴𝑦𝐵 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
4 xpsff1o.f . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
54fmpo 7760 . . . 4 (∀𝑥𝐴𝑦𝐵 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵))
63, 5mpbi 232 . . 3 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
7 1st2nd2 7722 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
87fveq2d 6669 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
9 df-ov 7153 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
10 xp1st 7715 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (1st𝑧) ∈ 𝐴)
11 xp2nd 7716 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (2nd𝑧) ∈ 𝐵)
124xpsfval 16833 . . . . . . . . 9 (((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
1310, 11, 12syl2anc 586 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
149, 13syl5eqr 2870 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
158, 14eqtrd 2856 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
16 1st2nd2 7722 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
1716fveq2d 6669 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
18 df-ov 7153 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
19 xp1st 7715 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (1st𝑤) ∈ 𝐴)
20 xp2nd 7716 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (2nd𝑤) ∈ 𝐵)
214xpsfval 16833 . . . . . . . . 9 (((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2219, 20, 21syl2anc 586 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2318, 22syl5eqr 2870 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2417, 23eqtrd 2856 . . . . . 6 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2515, 24eqeqan12d 2838 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) ↔ {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}))
26 fveq1 6664 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘∅) = ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘∅))
27 fvex 6678 . . . . . . . . 9 (1st𝑧) ∈ V
28 fvpr0o 16826 . . . . . . . . 9 ((1st𝑧) ∈ V → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘∅) = (1st𝑧))
2927, 28ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘∅) = (1st𝑧)
30 fvex 6678 . . . . . . . . 9 (1st𝑤) ∈ V
31 fvpr0o 16826 . . . . . . . . 9 ((1st𝑤) ∈ V → ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘∅) = (1st𝑤))
3230, 31ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘∅) = (1st𝑤)
3326, 29, 323eqtr3g 2879 . . . . . . 7 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → (1st𝑧) = (1st𝑤))
34 fveq1 6664 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘1o) = ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘1o))
35 fvex 6678 . . . . . . . . 9 (2nd𝑧) ∈ V
36 fvpr1o 16827 . . . . . . . . 9 ((2nd𝑧) ∈ V → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘1o) = (2nd𝑧))
3735, 36ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘1o) = (2nd𝑧)
38 fvex 6678 . . . . . . . . 9 (2nd𝑤) ∈ V
39 fvpr1o 16827 . . . . . . . . 9 ((2nd𝑤) ∈ V → ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘1o) = (2nd𝑤))
4038, 39ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘1o) = (2nd𝑤)
4134, 37, 403eqtr3g 2879 . . . . . . 7 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → (2nd𝑧) = (2nd𝑤))
4233, 41opeq12d 4805 . . . . . 6 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩)
437, 16eqeqan12d 2838 . . . . . 6 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
4442, 43syl5ibr 248 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → 𝑧 = 𝑤))
4525, 44sylbid 242 . . . 4 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
4645rgen2 3203 . . 3 𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
47 dff13 7007 . . 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 709 . 2 𝐹:(𝐴 × 𝐵)–1-1X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
49 xpsfrnel 16829 . . . . . 6 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2o ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵))
5049simp2bi 1142 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
5149simp3bi 1143 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1o) ∈ 𝐵)
524xpsfval 16833 . . . . . . 7 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1o, (𝑧‘1o)⟩})
5350, 51, 52syl2anc 586 . . . . . 6 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1o, (𝑧‘1o)⟩})
54 ixpfn 8461 . . . . . . 7 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2o)
55 xpsfeq 16830 . . . . . . 7 (𝑧 Fn 2o → {⟨∅, (𝑧‘∅)⟩, ⟨1o, (𝑧‘1o)⟩} = 𝑧)
5654, 55syl 17 . . . . . 6 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → {⟨∅, (𝑧‘∅)⟩, ⟨1o, (𝑧‘1o)⟩} = 𝑧)
5753, 56eqtr2d 2857 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1o)))
58 rspceov 7197 . . . . 5 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵𝑧 = ((𝑧‘∅)𝐹(𝑧‘1o))) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
5950, 51, 57, 58syl3anc 1367 . . . 4 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
6059rgen 3148 . . 3 𝑧X 𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏)
61 foov 7316 . . 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 709 . 2 𝐹:(𝐴 × 𝐵)–ontoX𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
63 df-f1o 6357 . 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 709 1 𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  Vcvv 3495  ∅c0 4291  ifcif 4467  {cpr 4563  ⟨cop 4567   × cxp 5548   Fn wfn 6345  ⟶wf 6346  –1-1→wf1 6347  –onto→wfo 6348  –1-1-onto→wf1o 6349  ‘cfv 6350  (class class class)co 7150   ∈ cmpo 7152  1st c1st 7681  2nd c2nd 7682  1oc1o 8089  2oc2o 8090  Xcixp 8455 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8283  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507 This theorem is referenced by:  xpsfrn  16835  xpsff1o2  16836
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