Theorem tposoprab 8248

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 8245	. 2 ⊢ tpos 𝐹 = tpos {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
3		reldmoprab 7510	. . 3 ⊢ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		dftpos3 8230	. . 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 2897	. . . . 5 ⊢ Ⅎ𝑦⟨𝑏, 𝑎⟩
7		nfoprab2 7467	. . . . 5 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
8		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑦𝑐
9	6, 7, 8	nfbr 5188	. . . 4 ⊢ Ⅎ𝑦⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
10		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑥⟨𝑏, 𝑎⟩
11		nfoprab1 7466	. . . . 5 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
12		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑥𝑐
13	10, 11, 12	nfbr 5188	. . . 4 ⊢ Ⅎ𝑥⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
14		nfv 1909	. . . 4 ⊢ Ⅎ𝑎⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
15		nfv 1909	. . . 4 ⊢ Ⅎ𝑏⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
16		opeq12 4870	. . . . . 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 5151	. . . 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 7494	. . 3 ⊢ {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑐⟩ ∣ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐}
20		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩
21		nfoprab3 7468	. . . . 5 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
22		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑧𝑐
23	20, 21, 22	nfbr 5188	. . . 4 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
24		nfv 1909	. . . 4 ⊢ Ⅎ𝑐𝜑
25		breq2 5145	. . . . 5 ⊢ (𝑐 = 𝑧 → (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐 ↔ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧))
26		df-br 5142	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
27		oprabidw 7436	. . . . . 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 7496	. . 3 ⊢ {⟨⟨𝑦, 𝑥⟩, 𝑐⟩ ∣ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
31	19, 30	eqtri 2754	. 2 ⊢ {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
32	2, 5, 31	3eqtri 2758	1 ⊢ tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ⟨cop 4629   class class class wbr 5141  dom cdm 5669  Rel wrel 5674  {coprab 7406  tpos ctpos 8211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-fv 6545  df-oprab 7409  df-tpos 8212

This theorem is referenced by:  tposmpo  8249