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Mirrors > Home > MPE Home > Th. List > xpsff1o2 | Structured version Visualization version GIF version |
Description: The function appearing in xpsval 16835 is a bijection from the cartesian product to the indexed cartesian product indexed on the pair 2o = {∅, 1o}. (Contributed by Mario Carneiro, 24-Jan-2015.) |
Ref | Expression |
---|---|
xpsff1o.f | ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) |
Ref | Expression |
---|---|
xpsff1o2 | ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsff1o.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | |
2 | 1 | xpsff1o 16832 | . 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) |
3 | f1of1 6589 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵)) | |
4 | f1f1orn 6601 | . 2 ⊢ (𝐹:(𝐴 × 𝐵)–1-1→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) → 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹) | |
5 | 2, 3, 4 | mp2b 10 | 1 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→ran 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∅c0 4243 ifcif 4425 {cpr 4527 〈cop 4531 × cxp 5517 ran crn 5520 –1-1→wf1 6321 –1-1-onto→wf1o 6323 ∈ cmpo 7137 1oc1o 8078 2oc2o 8079 Xcixp 8444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 |
This theorem is referenced by: xpsbas 16837 xpsaddlem 16838 xpsadd 16839 xpsmul 16840 xpssca 16841 xpsvsca 16842 xpsless 16843 xpsle 16844 xpsmnd 17943 xpsgrp 18210 xpstps 22415 xpstopnlem2 22416 xpsdsfn 22984 xpsxmet 22987 xpsdsval 22988 xpsmet 22989 xpsxms 23141 xpsms 23142 |
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