Theorem tposoprab 6333

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 6330	. 2 ⊢ tpos 𝐹 = tpos {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
3		reldmoprab 6003	. . 3 ⊢ Rel dom {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4		dftpos3 6315	. . 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 2336	. . . . 5 ⊢ Ⅎ𝑦⟨𝑏, 𝑎⟩
7		nfoprab2 5968	. . . . 5 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
8		nfcv 2336	. . . . 5 ⊢ Ⅎ𝑦𝑐
9	6, 7, 8	nfbr 4075	. . . 4 ⊢ Ⅎ𝑦⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
10		nfcv 2336	. . . . 5 ⊢ Ⅎ𝑥⟨𝑏, 𝑎⟩
11		nfoprab1 5967	. . . . 5 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
12		nfcv 2336	. . . . 5 ⊢ Ⅎ𝑥𝑐
13	10, 11, 12	nfbr 4075	. . . 4 ⊢ Ⅎ𝑥⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
14		nfv 1539	. . . 4 ⊢ Ⅎ𝑎⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
15		nfv 1539	. . . 4 ⊢ Ⅎ𝑏⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
16		opeq12 3806	. . . . . 6 ⊢ ((𝑏 = 𝑥 ∧ 𝑎 = 𝑦) → ⟨𝑏, 𝑎⟩ = ⟨𝑥, 𝑦⟩)
17	16	ancoms 268	. . . . 5 ⊢ ((𝑎 = 𝑦 ∧ 𝑏 = 𝑥) → ⟨𝑏, 𝑎⟩ = ⟨𝑥, 𝑦⟩)
18	17	breq1d 4039	. . . 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 5992	. . 3 ⊢ {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑐⟩ ∣ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐}
20		nfcv 2336	. . . . 5 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩
21		nfoprab3 5969	. . . . 5 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
22		nfcv 2336	. . . . 5 ⊢ Ⅎ𝑧𝑐
23	20, 21, 22	nfbr 4075	. . . 4 ⊢ Ⅎ𝑧⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐
24		nfv 1539	. . . 4 ⊢ Ⅎ𝑐𝜑
25		breq2 4033	. . . . 5 ⊢ (𝑐 = 𝑧 → (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐 ↔ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧))
26		df-br 4030	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧 ↔ ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
27		oprabid 5950	. . . . . 6 ⊢ (⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜑)
28	26, 27	bitri 184	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑧 ↔ 𝜑)
29	25, 28	bitrdi 196	. . . 4 ⊢ (𝑐 = 𝑧 → (⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐 ↔ 𝜑))
30	23, 24, 29	cbvoprab3 5994	. . 3 ⊢ {⟨⟨𝑦, 𝑥⟩, 𝑐⟩ ∣ ⟨𝑥, 𝑦⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
31	19, 30	eqtri 2214	. 2 ⊢ {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}𝑐} = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}
32	2, 5, 31	3eqtri 2218	1 ⊢ tpos 𝐹 = {⟨⟨𝑦, 𝑥⟩, 𝑧⟩ ∣ 𝜑}

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ⟨cop 3621   class class class wbr 4029  dom cdm 4659  Rel wrel 4664  {coprab 5919  tpos ctpos 6297

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-oprab 5922  df-tpos 6298

This theorem is referenced by:  tposmpo  6334