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 4509 | . . . . . 6 ⊢ if(𝑘 = ∅, 𝐴, 𝐴) = 𝐴 | |
2 | 1 | eleq2i 2907 | . . . . 5 ⊢ (({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ if(𝑘 = ∅, 𝐴, 𝐴) ↔ ({〈∅, 𝑋〉, 〈1o, 𝑌〉}‘𝑘) ∈ 𝐴) |
3 | 2 | ralbii 3168 | . . . 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 8468 | . . . 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 8269 | . . . . . . 7 ⊢ 2o ∈ ω | |
8 | fnex 6983 | . . . . . . 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 6885 | . . 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 16840 | . 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 1536 ∈ wcel 2113 ∀wral 3141 Vcvv 3497 ∅c0 4294 ifcif 4470 {cpr 4572 〈cop 4576 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 ωcom 7583 1oc1o 8098 2oc2o 8099 Xcixp 8464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 |
This theorem is referenced by: xpsmnd 17954 xpsgrp 18221 dmdprdpr 19174 dprdpr 19175 xpstopnlem1 22420 xpstps 22421 xpsxms 23147 xpsms 23148 |
Copyright terms: Public domain | W3C validator |