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Theorem xpsff1o 16828
Description: The function appearing in xpsval 16831 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 16825 . . . . . 6 ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥𝐴𝑦𝐵))
21biimpri 229 . . . . 5 ((𝑥𝐴𝑦𝐵) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵))
32rgen2 3200 . . . 4 𝑥𝐴𝑦𝐵 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
4 xpsff1o.f . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
54fmpo 7755 . . . 4 (∀𝑥𝐴𝑦𝐵 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵))
63, 5mpbi 231 . . 3 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
7 1st2nd2 7717 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
87fveq2d 6667 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
9 df-ov 7148 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
10 xp1st 7710 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (1st𝑧) ∈ 𝐴)
11 xp2nd 7711 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (2nd𝑧) ∈ 𝐵)
124xpsfval 16827 . . . . . . . . 9 (((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
1310, 11, 12syl2anc 584 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
149, 13syl5eqr 2867 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
158, 14eqtrd 2853 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
16 1st2nd2 7717 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
1716fveq2d 6667 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
18 df-ov 7148 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
19 xp1st 7710 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (1st𝑤) ∈ 𝐴)
20 xp2nd 7711 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (2nd𝑤) ∈ 𝐵)
214xpsfval 16827 . . . . . . . . 9 (((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2219, 20, 21syl2anc 584 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2318, 22syl5eqr 2867 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2417, 23eqtrd 2853 . . . . . 6 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2515, 24eqeqan12d 2835 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) ↔ {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}))
26 fveq1 6662 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘∅) = ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘∅))
27 fvex 6676 . . . . . . . . 9 (1st𝑧) ∈ V
28 fvpr0o 16820 . . . . . . . . 9 ((1st𝑧) ∈ V → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘∅) = (1st𝑧))
2927, 28ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘∅) = (1st𝑧)
30 fvex 6676 . . . . . . . . 9 (1st𝑤) ∈ V
31 fvpr0o 16820 . . . . . . . . 9 ((1st𝑤) ∈ V → ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘∅) = (1st𝑤))
3230, 31ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘∅) = (1st𝑤)
3326, 29, 323eqtr3g 2876 . . . . . . 7 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → (1st𝑧) = (1st𝑤))
34 fveq1 6662 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘1o) = ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘1o))
35 fvex 6676 . . . . . . . . 9 (2nd𝑧) ∈ V
36 fvpr1o 16821 . . . . . . . . 9 ((2nd𝑧) ∈ V → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘1o) = (2nd𝑧))
3735, 36ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘1o) = (2nd𝑧)
38 fvex 6676 . . . . . . . . 9 (2nd𝑤) ∈ V
39 fvpr1o 16821 . . . . . . . . 9 ((2nd𝑤) ∈ V → ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘1o) = (2nd𝑤))
4038, 39ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘1o) = (2nd𝑤)
4134, 37, 403eqtr3g 2876 . . . . . . 7 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → (2nd𝑧) = (2nd𝑤))
4233, 41opeq12d 4803 . . . . . 6 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩)
437, 16eqeqan12d 2835 . . . . . 6 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
4442, 43syl5ibr 247 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → 𝑧 = 𝑤))
4525, 44sylbid 241 . . . 4 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
4645rgen2 3200 . . 3 𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
47 dff13 7004 . . 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 707 . 2 𝐹:(𝐴 × 𝐵)–1-1X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
49 xpsfrnel 16823 . . . . . 6 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2o ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵))
5049simp2bi 1138 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
5149simp3bi 1139 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1o) ∈ 𝐵)
524xpsfval 16827 . . . . . . 7 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1o, (𝑧‘1o)⟩})
5350, 51, 52syl2anc 584 . . . . . 6 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1o, (𝑧‘1o)⟩})
54 ixpfn 8455 . . . . . . 7 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2o)
55 xpsfeq 16824 . . . . . . 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 2854 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1o)))
58 rspceov 7192 . . . . 5 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵𝑧 = ((𝑧‘∅)𝐹(𝑧‘1o))) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
5950, 51, 57, 58syl3anc 1363 . . . 4 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
6059rgen 3145 . . 3 𝑧X 𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏)
61 foov 7311 . . 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 707 . 2 𝐹:(𝐴 × 𝐵)–ontoX𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
63 df-f1o 6355 . 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 707 1 𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  c0 4288  ifcif 4463  {cpr 4559  cop 4563   × cxp 5546   Fn wfn 6343  wf 6344  1-1wf1 6345  ontowfo 6346  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  1oc1o 8084  2oc2o 8085  Xcixp 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-ixp 8450  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501
This theorem is referenced by:  xpsfrn  16829  xpsff1o2  16830
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