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Theorem resoprab 5539
 Description: Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)
Assertion
Ref Expression
resoprab ({⟨⟨x, y⟩, z⟩ ∣ φ} ↾ (A × B)) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) φ)}
Distinct variable groups:   x,y,z,A   x,B,y,z
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem resoprab
Dummy variable w is distinct from all other variables.
StepHypRef Expression
1 resopab 4595 . . 3 ({⟨w, z⟩ ∣ xy(w = ⟨x, y φ)} ↾ (A × B)) = {⟨w, z⟩ ∣ (w (A × B) xy(w = ⟨x, y φ))}
2 19.42vv 1785 . . . . 5 (xy(w (A × B) (w = ⟨x, y φ)) ↔ (w (A × B) xy(w = ⟨x, y φ)))
3 an12 495 . . . . . . 7 ((w (A × B) (w = ⟨x, y φ)) ↔ (w = ⟨x, y (w (A × B) φ)))
4 eleq1 2097 . . . . . . . . . 10 (w = ⟨x, y⟩ → (w (A × B) ↔ ⟨x, y (A × B)))
5 opelxp 4317 . . . . . . . . . 10 (⟨x, y (A × B) ↔ (x A y B))
64, 5syl6bb 185 . . . . . . . . 9 (w = ⟨x, y⟩ → (w (A × B) ↔ (x A y B)))
76anbi1d 438 . . . . . . . 8 (w = ⟨x, y⟩ → ((w (A × B) φ) ↔ ((x A y B) φ)))
87pm5.32i 427 . . . . . . 7 ((w = ⟨x, y (w (A × B) φ)) ↔ (w = ⟨x, y ((x A y B) φ)))
93, 8bitri 173 . . . . . 6 ((w (A × B) (w = ⟨x, y φ)) ↔ (w = ⟨x, y ((x A y B) φ)))
1092exbii 1494 . . . . 5 (xy(w (A × B) (w = ⟨x, y φ)) ↔ xy(w = ⟨x, y ((x A y B) φ)))
112, 10bitr3i 175 . . . 4 ((w (A × B) xy(w = ⟨x, y φ)) ↔ xy(w = ⟨x, y ((x A y B) φ)))
1211opabbii 3815 . . 3 {⟨w, z⟩ ∣ (w (A × B) xy(w = ⟨x, y φ))} = {⟨w, z⟩ ∣ xy(w = ⟨x, y ((x A y B) φ))}
131, 12eqtri 2057 . 2 ({⟨w, z⟩ ∣ xy(w = ⟨x, y φ)} ↾ (A × B)) = {⟨w, z⟩ ∣ xy(w = ⟨x, y ((x A y B) φ))}
14 dfoprab2 5494 . . 3 {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ xy(w = ⟨x, y φ)}
1514reseq1i 4551 . 2 ({⟨⟨x, y⟩, z⟩ ∣ φ} ↾ (A × B)) = ({⟨w, z⟩ ∣ xy(w = ⟨x, y φ)} ↾ (A × B))
16 dfoprab2 5494 . 2 {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) φ)} = {⟨w, z⟩ ∣ xy(w = ⟨x, y ((x A y B) φ))}
1713, 15, 163eqtr4i 2067 1 ({⟨⟨x, y⟩, z⟩ ∣ φ} ↾ (A × B)) = {⟨⟨x, y⟩, z⟩ ∣ ((x A y B) φ)}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ⟨cop 3370  {copab 3808   × cxp 4286   ↾ cres 4290  {coprab 5456 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-bndl 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-ral 2305  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-opab 3810  df-xp 4294  df-rel 4295  df-res 4300  df-oprab 5459 This theorem is referenced by:  resoprab2  5540
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