Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4506 | . . . . . 6 ⊢ if(𝑘 = ∅, 𝐴, 𝐴) = 𝐴 | |
2 | 1 | eleq2i 2904 | . . . . 5 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴) |
3 | 2 | ralbii 3165 | . . . 4 ⊢ (∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴) ↔ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴) |
4 | 3 | anbi2i 624 | . . 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 1085 | . . . 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 8459 | . . . 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 8260 | . . . . . . 7 ⊢ 2o ∈ ω | |
8 | fnex 6974 | . . . . . . 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 563 | . . . . 5 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ V ∧ {〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o)) |
11 | 10 | anbi1i 625 | . . . 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 305 | . . 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 6877 | . . 3 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴 ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} Fn 2o ∧ ∀𝑘 ∈ 2o ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴)) | |
14 | 4, 12, 13 | 3bitr4i 305 | . 2 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ {〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴) |
15 | xpsfrnel2 16831 | . 2 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉} ∈ X𝑘 ∈ 2o if(𝑘 = ∅, 𝐴, 𝐴) ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) | |
16 | 14, 15 | bitr3i 279 | 1 ⊢ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}:2o⟶𝐴 ↔ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3495 ∅c0 4291 ifcif 4467 {cpr 4563 〈cop 4567 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 ωcom 7574 1oc1o 8089 2oc2o 8090 Xcixp 8455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 |
This theorem is referenced by: xpsmnd 17945 xpsgrp 18212 dmdprdpr 19165 dprdpr 19166 xpstopnlem1 22411 xpstps 22412 xpsxms 23138 xpsms 23139 |
Copyright terms: Public domain | W3C validator |