Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sprval Structured version   Visualization version   GIF version

Theorem sprval 42576
 Description: The set of all unordered pairs over a given set 𝑉. (Contributed by AV, 21-Nov-2021.)
Assertion
Ref Expression
sprval (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
Distinct variable groups:   𝑉,𝑎,𝑏,𝑝   𝑊,𝑎,𝑏,𝑝

Proof of Theorem sprval
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 df-spr 42575 . . 3 Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}})
21a1i 11 . 2 (𝑉𝑊 → Pairs = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}}))
3 id 22 . . . . 5 (𝑣 = 𝑉𝑣 = 𝑉)
4 rexeq 3351 . . . . 5 (𝑣 = 𝑉 → (∃𝑏𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}))
53, 4rexeqbidv 3365 . . . 4 (𝑣 = 𝑉 → (∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}))
65adantl 475 . . 3 ((𝑉𝑊𝑣 = 𝑉) → (∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}))
76abbidv 2946 . 2 ((𝑉𝑊𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎𝑣𝑏𝑣 𝑝 = {𝑎, 𝑏}} = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
8 elex 3429 . 2 (𝑉𝑊𝑉 ∈ V)
9 zfpair2 5128 . . . . . . . 8 {𝑎, 𝑏} ∈ V
10 eueq 3602 . . . . . . . 8 ({𝑎, 𝑏} ∈ V ↔ ∃!𝑝 𝑝 = {𝑎, 𝑏})
119, 10mpbi 222 . . . . . . 7 ∃!𝑝 𝑝 = {𝑎, 𝑏}
12 euabex 5150 . . . . . . 7 (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1311, 12mp1i 13 . . . . . 6 (𝑉𝑊 → {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
1413ralrimivw 3176 . . . . 5 (𝑉𝑊 → ∀𝑏𝑉 {𝑝𝑝 = {𝑎, 𝑏}} ∈ V)
15 abrexex2g 7405 . . . . 5 ((𝑉𝑊 ∧ ∀𝑏𝑉 {𝑝𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1614, 15mpdan 680 . . . 4 (𝑉𝑊 → {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1716ralrimivw 3176 . . 3 (𝑉𝑊 → ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
18 abrexex2g 7405 . . 3 ((𝑉𝑊 ∧ ∀𝑎𝑉 {𝑝 ∣ ∃𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
1917, 18mpdan 680 . 2 (𝑉𝑊 → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ∈ V)
202, 7, 8, 19fvmptd 6535 1 (𝑉𝑊 → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∃!weu 2639  {cab 2811  ∀wral 3117  ∃wrex 3118  Vcvv 3414  {cpr 4399   ↦ cmpt 4952  ‘cfv 6123  Pairscspr 42574 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-spr 42575 This theorem is referenced by:  sprvalpw  42577  sprssspr  42578
 Copyright terms: Public domain W3C validator