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Theorem tposoprab 5836
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.
StepHypRef Expression
1 tposoprab.1 . . 3 𝐹 = {⟨⟨x, y⟩, z⟩ ∣ φ}
21tposeqi 5833 . 2 tpos 𝐹 = tpos {⟨⟨x, y⟩, z⟩ ∣ φ}
3 reldmoprab 5531 . . 3 Rel dom {⟨⟨x, y⟩, z⟩ ∣ φ}
4 dftpos3 5818 . . 3 (Rel dom {⟨⟨x, y⟩, z⟩ ∣ φ} → tpos {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐})
53, 4ax-mp 7 . 2 tpos {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐}
6 nfcv 2175 . . . . 5 y𝑏, 𝑎
7 nfoprab2 5497 . . . . 5 y{⟨⟨x, y⟩, z⟩ ∣ φ}
8 nfcv 2175 . . . . 5 y𝑐
96, 7, 8nfbr 3799 . . . 4 y𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐
10 nfcv 2175 . . . . 5 x𝑏, 𝑎
11 nfoprab1 5496 . . . . 5 x{⟨⟨x, y⟩, z⟩ ∣ φ}
12 nfcv 2175 . . . . 5 x𝑐
1310, 11, 12nfbr 3799 . . . 4 x𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐
14 nfv 1418 . . . 4 𝑎x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐
15 nfv 1418 . . . 4 𝑏x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐
16 opeq12 3542 . . . . . 6 ((𝑏 = x 𝑎 = y) → ⟨𝑏, 𝑎⟩ = ⟨x, y⟩)
1716ancoms 255 . . . . 5 ((𝑎 = y 𝑏 = x) → ⟨𝑏, 𝑎⟩ = ⟨x, y⟩)
1817breq1d 3765 . . . 4 ((𝑎 = y 𝑏 = x) → (⟨𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐 ↔ ⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐))
199, 13, 14, 15, 18cbvoprab12 5520 . . 3 {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐} = {⟨⟨y, x⟩, 𝑐⟩ ∣ ⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐}
20 nfcv 2175 . . . . 5 zx, y
21 nfoprab3 5498 . . . . 5 z{⟨⟨x, y⟩, z⟩ ∣ φ}
22 nfcv 2175 . . . . 5 z𝑐
2320, 21, 22nfbr 3799 . . . 4 zx, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐
24 nfv 1418 . . . 4 𝑐φ
25 breq2 3759 . . . . 5 (𝑐 = z → (⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐 ↔ ⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}z))
26 df-br 3756 . . . . . 6 (⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}z ↔ ⟨⟨x, y⟩, z {⟨⟨x, y⟩, z⟩ ∣ φ})
27 oprabid 5480 . . . . . 6 (⟨⟨x, y⟩, z {⟨⟨x, y⟩, z⟩ ∣ φ} ↔ φ)
2826, 27bitri 173 . . . . 5 (⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}zφ)
2925, 28syl6bb 185 . . . 4 (𝑐 = z → (⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐φ))
3023, 24, 29cbvoprab3 5522 . . 3 {⟨⟨y, x⟩, 𝑐⟩ ∣ ⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐} = {⟨⟨y, x⟩, z⟩ ∣ φ}
3119, 30eqtri 2057 . 2 {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐} = {⟨⟨y, x⟩, z⟩ ∣ φ}
322, 5, 313eqtri 2061 1 tpos 𝐹 = {⟨⟨y, x⟩, z⟩ ∣ φ}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  cop 3370   class class class wbr 3755  dom cdm 4288  Rel wrel 4293  {coprab 5456  tpos ctpos 5800
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853  df-oprab 5459  df-tpos 5801
This theorem is referenced by:  tposmpt2  5837
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