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Theorem tposoprab 5813
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 5810 . 2 tpos 𝐹 = tpos {⟨⟨x, y⟩, z⟩ ∣ φ}
3 reldmoprab 5508 . . 3 Rel dom {⟨⟨x, y⟩, z⟩ ∣ φ}
4 dftpos3 5795 . . 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 2156 . . . . 5 y𝑏, 𝑎
7 nfoprab2 5474 . . . . 5 y{⟨⟨x, y⟩, z⟩ ∣ φ}
8 nfcv 2156 . . . . 5 y𝑐
96, 7, 8nfbr 3778 . . . 4 y𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐
10 nfcv 2156 . . . . 5 x𝑏, 𝑎
11 nfoprab1 5473 . . . . 5 x{⟨⟨x, y⟩, z⟩ ∣ φ}
12 nfcv 2156 . . . . 5 x𝑐
1310, 11, 12nfbr 3778 . . . 4 x𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐
14 nfv 1398 . . . 4 𝑎x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐
15 nfv 1398 . . . 4 𝑏x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐
16 opeq12 3521 . . . . . 6 ((𝑏 = x 𝑎 = y) → ⟨𝑏, 𝑎⟩ = ⟨x, y⟩)
1716ancoms 255 . . . . 5 ((𝑎 = y 𝑏 = x) → ⟨𝑏, 𝑎⟩ = ⟨x, y⟩)
1817breq1d 3744 . . . 4 ((𝑎 = y 𝑏 = x) → (⟨𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐 ↔ ⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐))
199, 13, 14, 15, 18cbvoprab12 5497 . . 3 {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐} = {⟨⟨y, x⟩, 𝑐⟩ ∣ ⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐}
20 nfcv 2156 . . . . 5 zx, y
21 nfoprab3 5475 . . . . 5 z{⟨⟨x, y⟩, z⟩ ∣ φ}
22 nfcv 2156 . . . . 5 z𝑐
2320, 21, 22nfbr 3778 . . . 4 zx, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐
24 nfv 1398 . . . 4 𝑐φ
25 breq2 3738 . . . . 5 (𝑐 = z → (⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐 ↔ ⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}z))
26 df-br 3735 . . . . . 6 (⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}z ↔ ⟨⟨x, y⟩, z {⟨⟨x, y⟩, z⟩ ∣ φ})
27 oprabid 5457 . . . . . 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 5499 . . 3 {⟨⟨y, x⟩, 𝑐⟩ ∣ ⟨x, y⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐} = {⟨⟨y, x⟩, z⟩ ∣ φ}
3119, 30eqtri 2038 . 2 {⟨⟨𝑎, 𝑏⟩, 𝑐⟩ ∣ ⟨𝑏, 𝑎⟩{⟨⟨x, y⟩, z⟩ ∣ φ}𝑐} = {⟨⟨y, x⟩, z⟩ ∣ φ}
322, 5, 313eqtri 2042 1 tpos 𝐹 = {⟨⟨y, x⟩, z⟩ ∣ φ}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1226   wcel 1370  cop 3349   class class class wbr 3734  dom cdm 4268  Rel wrel 4273  {coprab 5433  tpos ctpos 5777
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-nul 3853  ax-pow 3897  ax-pr 3914  ax-un 4116  ax-setind 4200
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-sbc 2738  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-nul 3198  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833  df-oprab 5436  df-tpos 5778
This theorem is referenced by:  tposmpt2  5814
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