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Theorem iunopab 4009
Description: Move indexed union inside an ordered-pair abstraction. (Contributed by Stefan O'Rear, 20-Feb-2015.)
Assertion
Ref Expression
iunopab z A {⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ z A φ}
Distinct variable groups:   x,A   y,A   y,z   x,z
Allowed substitution hints:   φ(x,y,z)   A(z)

Proof of Theorem iunopab
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 elopab 3986 . . . . 5 (w {⟨x, y⟩ ∣ φ} ↔ xy(w = ⟨x, y φ))
21rexbii 2325 . . . 4 (z A w {⟨x, y⟩ ∣ φ} ↔ z A xy(w = ⟨x, y φ))
3 rexcom4 2571 . . . . 5 (z A xy(w = ⟨x, y φ) ↔ xz A y(w = ⟨x, y φ))
4 rexcom4 2571 . . . . . . 7 (z A y(w = ⟨x, y φ) ↔ yz A (w = ⟨x, y φ))
5 r19.42v 2461 . . . . . . . 8 (z A (w = ⟨x, y φ) ↔ (w = ⟨x, y z A φ))
65exbii 1493 . . . . . . 7 (yz A (w = ⟨x, y φ) ↔ y(w = ⟨x, y z A φ))
74, 6bitri 173 . . . . . 6 (z A y(w = ⟨x, y φ) ↔ y(w = ⟨x, y z A φ))
87exbii 1493 . . . . 5 (xz A y(w = ⟨x, y φ) ↔ xy(w = ⟨x, y z A φ))
93, 8bitri 173 . . . 4 (z A xy(w = ⟨x, y φ) ↔ xy(w = ⟨x, y z A φ))
102, 9bitri 173 . . 3 (z A w {⟨x, y⟩ ∣ φ} ↔ xy(w = ⟨x, y z A φ))
1110abbii 2150 . 2 {wz A w {⟨x, y⟩ ∣ φ}} = {wxy(w = ⟨x, y z A φ)}
12 df-iun 3650 . 2 z A {⟨x, y⟩ ∣ φ} = {wz A w {⟨x, y⟩ ∣ φ}}
13 df-opab 3810 . 2 {⟨x, y⟩ ∣ z A φ} = {wxy(w = ⟨x, y z A φ)}
1411, 12, 133eqtr4i 2067 1 z A {⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ z A φ}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wrex 2301  cop 3370   ciun 3648  {copab 3808
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-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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-opab 3810
This theorem is referenced by: (None)
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