Theorem tposoprab 8202

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 8199	. 2 ⊢ tpos 𝐹 = tpos {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
3		reldmoprab 7463	. . 3 ⊢ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		dftpos3 8184	. . 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 2901	. . . . 5 ⊢ Ⅎ𝑦⟨𝑏, 𝑎⟩
7		nfoprab2 7418	. . . . 5 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
8		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑦𝑐
9	6, 7, 8	nfbr 5119	. . . 4 ⊢ Ⅎ𝑦⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
10		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑥⟨𝑏, 𝑎⟩
11		nfoprab1 7417	. . . . 5 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
12		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑥𝑐
13	10, 11, 12	nfbr 5119	. . . 4 ⊢ Ⅎ𝑥⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
14		nfv 1921	. . . 4 ⊢ Ⅎ𝑎⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
15		nfv 1921	. . . 4 ⊢ Ⅎ𝑏⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
16		opeq12 4806	. . . . . 6 ⊢ ((𝑏 = 𝑥 ∧ 𝑎 = 𝑦) → ⟨𝑏, 𝑎⟩ = ⟨𝑥, 𝑦⟩)
17	16	ancoms 459	. . . . 5 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → ⟨𝑏, 𝑎⟩ = ⟨𝑥, 𝑦⟩)
18	17	breq1d 5082	. . . 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 7445	. . 3 ⊢ {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑐⟩ ∣ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐}
20		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩
21		nfoprab3 7419	. . . . 5 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
22		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑧𝑐
23	20, 21, 22	nfbr 5119	. . . 4 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
24		nfv 1921	. . . 4 ⊢ Ⅎ𝑐𝜑
25		breq2 5076	. . . . 5 ⊢ (𝑐 = 𝑧 → (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐 ↔ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧))
26		df-br 5073	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
27		oprabidw 7387	. . . . . 6 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
28	26, 27	bitri 276	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧 ↔ 𝜑)
29	25, 28	bitrdi 288	. . . 4 ⊢ (𝑐 = 𝑧 → (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐 ↔ 𝜑))
30	23, 24, 29	cbvoprab3 7447	. . 3 ⊢ {⟨⟨𝑦, 𝑥⟩, 𝑐⟩ ∣ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
31	19, 30	eqtri 2762	. 2 ⊢ {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
32	2, 5, 31	3eqtri 2766	1 ⊢ tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ⟨cop 4561   class class class wbr 5072  dom cdm 5618  Rel wrel 5623  {coprab 7357  tpos ctpos 8165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-fv 6493  df-oprab 7360  df-tpos 8166

This theorem is referenced by:  tposmpo  8203