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Mirrors > Home > MPE Home > Th. List > xpsff1o2 | Structured version Visualization version GIF version |
Description: The function appearing in xpsval 16843 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 16840 | . 2 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐵) |
3 | f1of1 6614 | . 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 6626 | . 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 1537 ∅c0 4291 ifcif 4467 {cpr 4569 〈cop 4573 × cxp 5553 ran crn 5556 –1-1→wf1 6352 –1-1-onto→wf1o 6354 ∈ cmpo 7158 1oc1o 8095 2oc2o 8096 Xcixp 8461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 |
This theorem is referenced by: xpsbas 16845 xpsaddlem 16846 xpsadd 16847 xpsmul 16848 xpssca 16849 xpsvsca 16850 xpsless 16851 xpsle 16852 xpsmnd 17951 xpsgrp 18218 xpstps 22418 xpstopnlem2 22419 xpsdsfn 22987 xpsxmet 22990 xpsdsval 22991 xpsmet 22992 xpsxms 23144 xpsms 23145 |
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