Theorem tposoprab 7911

Description: Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)

Hypothesis

Ref	Expression
tposoprab.1	⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}

Assertion

Ref	Expression
tposoprab	⊢ tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}

Distinct variable group:   𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐹(𝑥,𝑦,𝑧)

Proof of Theorem tposoprab

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tposoprab.1	. . 3 ⊢ 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
2	1	tposeqi 7908	. 2 ⊢ tpos 𝐹 = tpos {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
3		reldmoprab 7238	. . 3 ⊢ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		dftpos3 7893	. . 3 ⊢ (Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → tpos {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐})
5	3, 4	ax-mp 5	. 2 ⊢ tpos {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐}
6		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑦⟨𝑏, 𝑎⟩
7		nfoprab2 7195	. . . . 5 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
8		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑦𝑐
9	6, 7, 8	nfbr 5077	. . . 4 ⊢ Ⅎ𝑦⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
10		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑥⟨𝑏, 𝑎⟩
11		nfoprab1 7194	. . . . 5 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
12		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑥𝑐
13	10, 11, 12	nfbr 5077	. . . 4 ⊢ Ⅎ𝑥⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
14		nfv 1915	. . . 4 ⊢ Ⅎ𝑎⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
15		nfv 1915	. . . 4 ⊢ Ⅎ𝑏⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
16		opeq12 4767	. . . . . 6 ⊢ ((𝑏 = 𝑥 ∧ 𝑎 = 𝑦) → ⟨𝑏, 𝑎⟩ = ⟨𝑥, 𝑦⟩)
17	16	ancoms 462	. . . . 5 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → ⟨𝑏, 𝑎⟩ = ⟨𝑥, 𝑦⟩)
18	17	breq1d 5040	. . . 4 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → (⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐 ↔ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐))
19	9, 13, 14, 15, 18	cbvoprab12 7222	. . 3 ⊢ {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑐⟩ ∣ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐}
20		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩
21		nfoprab3 7196	. . . . 5 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
22		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑧𝑐
23	20, 21, 22	nfbr 5077	. . . 4 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
24		nfv 1915	. . . 4 ⊢ Ⅎ𝑐𝜑
25		breq2 5034	. . . . 5 ⊢ (𝑐 = 𝑧 → (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐 ↔ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧))
26		df-br 5031	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
27		oprabidw 7166	. . . . . 6 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
28	26, 27	bitri 278	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧 ↔ 𝜑)
29	25, 28	syl6bb 290	. . . 4 ⊢ (𝑐 = 𝑧 → (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐 ↔ 𝜑))
30	23, 24, 29	cbvoprab3 7224	. . 3 ⊢ {⟨⟨𝑦, 𝑥⟩, 𝑐⟩ ∣ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
31	19, 30	eqtri 2821	. 2 ⊢ {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
32	2, 5, 31	3eqtri 2825	1 ⊢ tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   class class class wbr 5030  dom cdm 5519  Rel wrel 5524  {coprab 7136  tpos ctpos 7874

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-sep 5167  ax-nul 5174  ax-pow 5231  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-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  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-fv 6332  df-oprab 7139  df-tpos 7875

This theorem is referenced by:  tposmpo  7912