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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 4567 | . . . . . 6 ⊢ if(𝑘 = ∅, 𝐴, 𝐴) = 𝐴 | |
2 | 1 | eleq2i 2825 | . . . . 5 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴) |
3 | 2 | ralbii 3093 | . . . 4 ⊢ (∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴) ↔ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴) |
4 | 3 | anbi2i 623 | . . 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 1089 | . . . 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 8891 | . . . 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 8637 | . . . . . . 7 ⊢ 2o ∈ ω | |
8 | fnex 7215 | . . . . . . 7 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ 2o ∈ ω) → {〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V) | |
9 | 7, 8 | mpan2 689 | . . . . . 6 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o → {〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V) |
10 | 9 | pm4.71ri 561 | . . . . 5 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o)) |
11 | 10 | anbi1i 624 | . . . 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 302 | . . 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 7114 | . . 3 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴 ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴)) | |
14 | 4, 12, 13 | 3bitr4i 302 | . 2 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ {〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴) |
15 | xpsfrnel2 17506 | . 2 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) | |
16 | 14, 15 | bitr3i 276 | 1 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ∅c0 4321 ifcif 4527 {cpr 4629 〈cop 4633 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 ωcom 7851 1oc1o 8455 2oc2o 8456 Xcixp 8887 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-om 7852 df-1o 8462 df-2o 8463 df-ixp 8888 df-en 8936 df-fin 8939 |
This theorem is referenced by: xpsmnd 18661 xpsgrp 18938 dmdprdpr 19913 dprdpr 19914 xpsringd 20138 xpstopnlem1 23304 xpstps 23305 xpsxms 24034 xpsms 24035 xpsrngd 46666 |
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