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Theorem xpsff1o 17556
Description: The function appearing in xpsval 17559 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 17553 . . . . . 6 ({⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥𝐴𝑦𝐵))
21biimpri 227 . . . . 5 ((𝑥𝐴𝑦𝐵) → {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵))
32rgen2 3195 . . . 4 𝑥𝐴𝑦𝐵 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
4 xpsff1o.f . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
54fmpo 8078 . . . 4 (∀𝑥𝐴𝑦𝐵 {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵))
63, 5mpbi 229 . . 3 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
7 1st2nd2 8038 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
87fveq2d 6906 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
9 df-ov 7429 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
10 xp1st 8031 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (1st𝑧) ∈ 𝐴)
11 xp2nd 8032 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (2nd𝑧) ∈ 𝐵)
124xpsfval 17555 . . . . . . . . 9 (((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
1310, 11, 12syl2anc 582 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
149, 13eqtr3id 2782 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
158, 14eqtrd 2768 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩})
16 1st2nd2 8038 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
1716fveq2d 6906 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
18 df-ov 7429 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
19 xp1st 8031 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (1st𝑤) ∈ 𝐴)
20 xp2nd 8032 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (2nd𝑤) ∈ 𝐵)
214xpsfval 17555 . . . . . . . . 9 (((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2219, 20, 21syl2anc 582 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2318, 22eqtr3id 2782 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2417, 23eqtrd 2768 . . . . . 6 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩})
2515, 24eqeqan12d 2742 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) ↔ {⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}))
26 fveq1 6901 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘∅) = ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘∅))
27 fvex 6915 . . . . . . . . 9 (1st𝑧) ∈ V
28 fvpr0o 17548 . . . . . . . . 9 ((1st𝑧) ∈ V → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘∅) = (1st𝑧))
2927, 28ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘∅) = (1st𝑧)
30 fvex 6915 . . . . . . . . 9 (1st𝑤) ∈ V
31 fvpr0o 17548 . . . . . . . . 9 ((1st𝑤) ∈ V → ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘∅) = (1st𝑤))
3230, 31ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘∅) = (1st𝑤)
3326, 29, 323eqtr3g 2791 . . . . . . 7 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → (1st𝑧) = (1st𝑤))
34 fveq1 6901 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘1o) = ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘1o))
35 fvex 6915 . . . . . . . . 9 (2nd𝑧) ∈ V
36 fvpr1o 17549 . . . . . . . . 9 ((2nd𝑧) ∈ V → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘1o) = (2nd𝑧))
3735, 36ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩}‘1o) = (2nd𝑧)
38 fvex 6915 . . . . . . . . 9 (2nd𝑤) ∈ V
39 fvpr1o 17549 . . . . . . . . 9 ((2nd𝑤) ∈ V → ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘1o) = (2nd𝑤))
4038, 39ax-mp 5 . . . . . . . 8 ({⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩}‘1o) = (2nd𝑤)
4134, 37, 403eqtr3g 2791 . . . . . . 7 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → (2nd𝑧) = (2nd𝑤))
4233, 41opeq12d 4886 . . . . . 6 ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩)
437, 16eqeqan12d 2742 . . . . . 6 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
4442, 43imbitrrid 245 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ({⟨∅, (1st𝑧)⟩, ⟨1o, (2nd𝑧)⟩} = {⟨∅, (1st𝑤)⟩, ⟨1o, (2nd𝑤)⟩} → 𝑧 = 𝑤))
4525, 44sylbid 239 . . . 4 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
4645rgen2 3195 . . 3 𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
47 dff13 7271 . . 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 17551 . . . . . 6 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2o ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵))
5049simp2bi 1143 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
5149simp3bi 1144 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1o) ∈ 𝐵)
524xpsfval 17555 . . . . . . 7 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1o, (𝑧‘1o)⟩})
5350, 51, 52syl2anc 582 . . . . . 6 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1o)) = {⟨∅, (𝑧‘∅)⟩, ⟨1o, (𝑧‘1o)⟩})
54 ixpfn 8928 . . . . . . 7 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2o)
55 xpsfeq 17552 . . . . . . 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 2769 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1o)))
58 rspceov 7473 . . . . 5 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵𝑧 = ((𝑧‘∅)𝐹(𝑧‘1o))) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
5950, 51, 57, 58syl3anc 1368 . . . 4 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
6059rgen 3060 . . 3 𝑧X 𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏)
61 foov 7601 . . 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 6560 . 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 394   = wceq 1533  wcel 2098  wral 3058  wrex 3067  Vcvv 3473  c0 4326  ifcif 4532  {cpr 4634  cop 4638   × cxp 5680   Fn wfn 6548  wf 6549  1-1wf1 6550  ontowfo 6551  1-1-ontowf1o 6552  cfv 6553  (class class class)co 7426  cmpo 7428  1st c1st 7997  2nd c2nd 7998  1oc1o 8486  2oc2o 8487  Xcixp 8922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-1o 8493  df-2o 8494  df-ixp 8923  df-en 8971  df-fin 8974
This theorem is referenced by:  xpsfrn  17557  xpsff1o2  17558
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