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Mirrors > Home > MPE Home > Th. List > xpscf | Structured version Visualization version GIF version |
Description: Equivalent condition for the pair function to be a proper function on 𝐴. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xpscf | ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifid 4464 | . . . . . 6 ⊢ if(𝑘 = ∅, 𝐴, 𝐴) = 𝐴 | |
2 | 1 | eleq2i 2881 | . . . . 5 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴) |
3 | 2 | ralbii 3133 | . . . 4 ⊢ (∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴) ↔ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴) |
4 | 3 | anbi2i 625 | . . 3 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴)) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴)) |
5 | df-3an 1086 | . . . 4 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴)) ↔ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o) ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴))) | |
6 | elixp2 8448 | . . . 4 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴))) | |
7 | 2onn 8249 | . . . . . . 7 ⊢ 2o ∈ ω | |
8 | fnex 6957 | . . . . . . 7 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ 2o ∈ ω) → {〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V) | |
9 | 7, 8 | mpan2 690 | . . . . . 6 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o → {〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V) |
10 | 9 | pm4.71ri 564 | . . . . 5 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o)) |
11 | 10 | anbi1i 626 | . . . 4 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴)) ↔ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o) ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴))) |
12 | 5, 6, 11 | 3bitr4i 306 | . . 3 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴))) |
13 | ffnfv 6859 | . . 3 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴 ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴)) | |
14 | 4, 12, 13 | 3bitr4i 306 | . 2 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ {〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴) |
15 | xpsfrnel2 16829 | . 2 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) | |
16 | 14, 15 | bitr3i 280 | 1 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∅c0 4243 ifcif 4425 {cpr 4527 〈cop 4531 Fn wfn 6319 ⟶wf 6320 ‘cfv 6324 ωcom 7560 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-rep 5154 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-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: xpsmnd 17943 xpsgrp 18210 dmdprdpr 19164 dprdpr 19165 xpstopnlem1 22414 xpstps 22415 xpsxms 23141 xpsms 23142 |
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