Theorem tposoprab 8241

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 8238	. 2 ⊢ tpos 𝐹 = tpos {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
3		reldmoprab 7496	. . 3 ⊢ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		dftpos3 8223	. . 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 2891	. . . . 5 ⊢ Ⅎ𝑦⟨𝑏, 𝑎⟩
7		nfoprab2 7451	. . . . 5 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
8		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑦𝑐
9	6, 7, 8	nfbr 5154	. . . 4 ⊢ Ⅎ𝑦⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
10		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥⟨𝑏, 𝑎⟩
11		nfoprab1 7450	. . . . 5 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
12		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑥𝑐
13	10, 11, 12	nfbr 5154	. . . 4 ⊢ Ⅎ𝑥⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
14		nfv 1914	. . . 4 ⊢ Ⅎ𝑎⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
15		nfv 1914	. . . 4 ⊢ Ⅎ𝑏⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
16		opeq12 4839	. . . . . 6 ⊢ ((𝑏 = 𝑥 ∧ 𝑎 = 𝑦) → ⟨𝑏, 𝑎⟩ = ⟨𝑥, 𝑦⟩)
17	16	ancoms 458	. . . . 5 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → ⟨𝑏, 𝑎⟩ = ⟨𝑥, 𝑦⟩)
18	17	breq1d 5117	. . . 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 7478	. . 3 ⊢ {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑐⟩ ∣ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐}
20		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩
21		nfoprab3 7452	. . . . 5 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
22		nfcv 2891	. . . . 5 ⊢ Ⅎ𝑧𝑐
23	20, 21, 22	nfbr 5154	. . . 4 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
24		nfv 1914	. . . 4 ⊢ Ⅎ𝑐𝜑
25		breq2 5111	. . . . 5 ⊢ (𝑐 = 𝑧 → (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐 ↔ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧))
26		df-br 5108	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
27		oprabidw 7418	. . . . . 6 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
28	26, 27	bitri 275	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧 ↔ 𝜑)
29	25, 28	bitrdi 287	. . . 4 ⊢ (𝑐 = 𝑧 → (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐 ↔ 𝜑))
30	23, 24, 29	cbvoprab3 7480	. . 3 ⊢ {⟨⟨𝑦, 𝑥⟩, 𝑐⟩ ∣ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
31	19, 30	eqtri 2752	. 2 ⊢ {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
32	2, 5, 31	3eqtri 2756	1 ⊢ tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4595   class class class wbr 5107  dom cdm 5638  Rel wrel 5643  {coprab 7388  tpos ctpos 8204

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519  df-oprab 7391  df-tpos 8205

This theorem is referenced by:  tposmpo  8242