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Theorem xpsff1o 16614
 Description: The function appearing in xpsval 16618 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 16611 . . . . . 6 (({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑥𝐴𝑦𝐵))
21biimpri 220 . . . . 5 ((𝑥𝐴𝑦𝐵) → ({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵))
32rgen2 3157 . . . 4 𝑥𝐴𝑦𝐵 ({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
4 xpsff1o.f . . . . 5 𝐹 = (𝑥𝐴, 𝑦𝐵({𝑥} +𝑐 {𝑦}))
54fmpt2 7517 . . . 4 (∀𝑥𝐴𝑦𝐵 ({𝑥} +𝑐 {𝑦}) ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵))
63, 5mpbi 222 . . 3 𝐹:(𝐴 × 𝐵)⟶X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
7 1st2nd2 7484 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
87fveq2d 6450 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩))
9 df-ov 6925 . . . . . . . 8 ((1st𝑧)𝐹(2nd𝑧)) = (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩)
10 xp1st 7477 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (1st𝑧) ∈ 𝐴)
11 xp2nd 7478 . . . . . . . . 9 (𝑧 ∈ (𝐴 × 𝐵) → (2nd𝑧) ∈ 𝐵)
124xpsfval 16613 . . . . . . . . 9 (((1st𝑧) ∈ 𝐴 ∧ (2nd𝑧) ∈ 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
1310, 11, 12syl2anc 579 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐵) → ((1st𝑧)𝐹(2nd𝑧)) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
149, 13syl5eqr 2828 . . . . . . 7 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑧), (2nd𝑧)⟩) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
158, 14eqtrd 2814 . . . . . 6 (𝑧 ∈ (𝐴 × 𝐵) → (𝐹𝑧) = ({(1st𝑧)} +𝑐 {(2nd𝑧)}))
16 1st2nd2 7484 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → 𝑤 = ⟨(1st𝑤), (2nd𝑤)⟩)
1716fveq2d 6450 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩))
18 df-ov 6925 . . . . . . . 8 ((1st𝑤)𝐹(2nd𝑤)) = (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩)
19 xp1st 7477 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (1st𝑤) ∈ 𝐴)
20 xp2nd 7478 . . . . . . . . 9 (𝑤 ∈ (𝐴 × 𝐵) → (2nd𝑤) ∈ 𝐵)
214xpsfval 16613 . . . . . . . . 9 (((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2219, 20, 21syl2anc 579 . . . . . . . 8 (𝑤 ∈ (𝐴 × 𝐵) → ((1st𝑤)𝐹(2nd𝑤)) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2318, 22syl5eqr 2828 . . . . . . 7 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹‘⟨(1st𝑤), (2nd𝑤)⟩) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2417, 23eqtrd 2814 . . . . . 6 (𝑤 ∈ (𝐴 × 𝐵) → (𝐹𝑤) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}))
2515, 24eqeqan12d 2794 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) ↔ ({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)})))
26 fveq1 6445 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅))
27 fvex 6459 . . . . . . . . 9 (1st𝑧) ∈ V
28 xpsc0 16606 . . . . . . . . 9 ((1st𝑧) ∈ V → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (1st𝑧))
2927, 28ax-mp 5 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘∅) = (1st𝑧)
30 fvex 6459 . . . . . . . . 9 (1st𝑤) ∈ V
31 xpsc0 16606 . . . . . . . . 9 ((1st𝑤) ∈ V → (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅) = (1st𝑤))
3230, 31ax-mp 5 . . . . . . . 8 (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘∅) = (1st𝑤)
3326, 29, 323eqtr3g 2837 . . . . . . 7 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (1st𝑧) = (1st𝑤))
34 fveq1 6445 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1o) = (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1o))
35 fvex 6459 . . . . . . . . 9 (2nd𝑧) ∈ V
36 xpsc1 16607 . . . . . . . . 9 ((2nd𝑧) ∈ V → (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1o) = (2nd𝑧))
3735, 36ax-mp 5 . . . . . . . 8 (({(1st𝑧)} +𝑐 {(2nd𝑧)})‘1o) = (2nd𝑧)
38 fvex 6459 . . . . . . . . 9 (2nd𝑤) ∈ V
39 xpsc1 16607 . . . . . . . . 9 ((2nd𝑤) ∈ V → (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1o) = (2nd𝑤))
4038, 39ax-mp 5 . . . . . . . 8 (({(1st𝑤)} +𝑐 {(2nd𝑤)})‘1o) = (2nd𝑤)
4134, 37, 403eqtr3g 2837 . . . . . . 7 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → (2nd𝑧) = (2nd𝑤))
4233, 41opeq12d 4644 . . . . . 6 (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩)
437, 16eqeqan12d 2794 . . . . . 6 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (𝑧 = 𝑤 ↔ ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨(1st𝑤), (2nd𝑤)⟩))
4442, 43syl5ibr 238 . . . . 5 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → (({(1st𝑧)} +𝑐 {(2nd𝑧)}) = ({(1st𝑤)} +𝑐 {(2nd𝑤)}) → 𝑧 = 𝑤))
4525, 44sylbid 232 . . . 4 ((𝑧 ∈ (𝐴 × 𝐵) ∧ 𝑤 ∈ (𝐴 × 𝐵)) → ((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤))
4645rgen2 3157 . . 3 𝑧 ∈ (𝐴 × 𝐵)∀𝑤 ∈ (𝐴 × 𝐵)((𝐹𝑧) = (𝐹𝑤) → 𝑧 = 𝑤)
47 dff13 6784 . . 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 701 . 2 𝐹:(𝐴 × 𝐵)–1-1X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
49 xpsfrnel 16609 . . . . . 6 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) ↔ (𝑧 Fn 2o ∧ (𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵))
5049simp2bi 1137 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘∅) ∈ 𝐴)
5149simp3bi 1138 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → (𝑧‘1o) ∈ 𝐵)
524xpsfval 16613 . . . . . . 7 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1o)) = ({(𝑧‘∅)} +𝑐 {(𝑧‘1o)}))
5350, 51, 52syl2anc 579 . . . . . 6 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → ((𝑧‘∅)𝐹(𝑧‘1o)) = ({(𝑧‘∅)} +𝑐 {(𝑧‘1o)}))
54 ixpfn 8200 . . . . . . 7 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 Fn 2o)
55 xpsfeq 16610 . . . . . . 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 2815 . . . . 5 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝑧 = ((𝑧‘∅)𝐹(𝑧‘1o)))
58 rspceov 6968 . . . . 5 (((𝑧‘∅) ∈ 𝐴 ∧ (𝑧‘1o) ∈ 𝐵𝑧 = ((𝑧‘∅)𝐹(𝑧‘1o))) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
5950, 51, 57, 58syl3anc 1439 . . . 4 (𝑧X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → ∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏))
6059rgen 3104 . . 3 𝑧X 𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)∃𝑎𝐴𝑏𝐵 𝑧 = (𝑎𝐹𝑏)
61 foov 7085 . . 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 701 . 2 𝐹:(𝐴 × 𝐵)–ontoX𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
63 df-f1o 6142 . 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 701 1 𝐹:(𝐴 × 𝐵)–1-1-ontoX𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091  Vcvv 3398  ∅c0 4141  ifcif 4307  {csn 4398  ⟨cop 4404   × cxp 5353  ◡ccnv 5354   Fn wfn 6130  ⟶wf 6131  –1-1→wf1 6132  –onto→wfo 6133  –1-1-onto→wf1o 6134  ‘cfv 6135  (class class class)co 6922   ↦ cmpt2 6924  1st c1st 7443  2nd c2nd 7444  1oc1o 7836  2oc2o 7837  Xcixp 8194   +𝑐 ccda 9324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-er 8026  df-ixp 8195  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-cda 9325 This theorem is referenced by:  xpsfrn  16615  xpsff1o2  16617
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