Theorem prprval 44031

Description: The set of all proper unordered pairs over a given set 𝑉. (Contributed by AV, 29-Apr-2023.)

Assertion

Ref	Expression
prprval	⊢ (𝑉 ∈ 𝑊 → (Pairsproper‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})

Distinct variable groups:   𝑉,𝑎,𝑏,𝑝   𝑊,𝑎,𝑏,𝑝

Proof of Theorem prprval

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prpr 44030	. 2 ⊢ Pairsproper = (𝑣 ∈ V ↦ {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
2		rexeq 3359	. . . . 5 ⊢ (𝑣 = 𝑉 → (∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})))
3	2	rexeqbi1dv 3357	. . . 4 ⊢ (𝑣 = 𝑉 → (∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})))
4	3	abbidv 2862	. . 3 ⊢ (𝑣 = 𝑉 → {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
5	4	adantl 485	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑣 = 𝑉) → {𝑝 ∣ ∃𝑎 ∈ 𝑣 ∃𝑏 ∈ 𝑣 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})
6		elex 3459	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝑉 ∈ V)
7		simpr 488	. . . . . . . 8 ⊢ ((𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏}) → 𝑝 = {𝑎, 𝑏})
8	7	ss2abi 3994	. . . . . . 7 ⊢ {𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ⊆ {𝑝 ∣ 𝑝 = {𝑎, 𝑏}}
9		zfpair2 5296	. . . . . . . . 9 ⊢ {𝑎, 𝑏} ∈ V
10	9	eueqi 3648	. . . . . . . 8 ⊢ ∃!𝑝 𝑝 = {𝑎, 𝑏}
11		euabex 5318	. . . . . . . 8 ⊢ (∃!𝑝 𝑝 = {𝑎, 𝑏} → {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
12	10, 11	mp1i 13	. . . . . . 7 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V)
13		ssexg 5191	. . . . . . 7 ⊢ (({𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ⊆ {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∧ {𝑝 ∣ 𝑝 = {𝑎, 𝑏}} ∈ V) → {𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
14	8, 12, 13	sylancr 590	. . . . . 6 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
15	14	ralrimivw 3150	. . . . 5 ⊢ (𝑉 ∈ 𝑊 → ∀𝑏 ∈ 𝑉 {𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
16		abrexex2g 7647	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ ∀𝑏 ∈ 𝑉 {𝑝 ∣ (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V) → {𝑝 ∣ ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
17	15, 16	mpdan 686	. . . 4 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
18	17	ralrimivw 3150	. . 3 ⊢ (𝑉 ∈ 𝑊 → ∀𝑎 ∈ 𝑉 {𝑝 ∣ ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
19		abrexex2g 7647	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ ∀𝑎 ∈ 𝑉 {𝑝 ∣ ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V) → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
20	18, 19	mpdan 686	. 2 ⊢ (𝑉 ∈ 𝑊 → {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})} ∈ V)
21	1, 5, 6, 20	fvmptd2 6753	1 ⊢ (𝑉 ∈ 𝑊 → (Pairsproper‘𝑉) = {𝑝 ∣ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 (𝑎 ≠ 𝑏 ∧ 𝑝 = {𝑎, 𝑏})})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃!weu 2628  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  {cpr 4527  ‘cfv 6324  Pairs_propercprpr 44029

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-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  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-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  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-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-prpr 44030

This theorem is referenced by:  prprvalpw  44032  prprspr2  44035