Theorem dmoprab 5527

Description: The domain of an operation class abstraction. (Contributed by NM, 17-Mar-1995.) (Revised by David Abernethy, 19-Jun-2012.)

Assertion

Ref	Expression
dmoprab	⊢ dom {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨x, y⟩ ∣ ∃zφ}

Distinct variable groups:   x,z   y,z

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem dmoprab

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfoprab2 5494	. . 3 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
2	1	dmeqi 4479	. 2 ⊢ dom {⟨⟨x, y⟩, z⟩ ∣ φ} = dom {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
3		dmopab 4489	. 2 ⊢ dom {⟨w, z⟩ ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {w ∣ ∃z∃x∃y(w = ⟨x, y⟩ ∧ φ)}
4		exrot3 1577	. . . . 5 ⊢ (∃z∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃x∃y∃z(w = ⟨x, y⟩ ∧ φ))
5		19.42v 1783	. . . . . 6 ⊢ (∃z(w = ⟨x, y⟩ ∧ φ) ↔ (w = ⟨x, y⟩ ∧ ∃zφ))
6	5	2exbii 1494	. . . . 5 ⊢ (∃x∃y∃z(w = ⟨x, y⟩ ∧ φ) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ ∃zφ))
7	4, 6	bitri 173	. . . 4 ⊢ (∃z∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃x∃y(w = ⟨x, y⟩ ∧ ∃zφ))
8	7	abbii 2150	. . 3 ⊢ {w ∣ ∃z∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ ∃zφ)}
9		df-opab 3810	. . 3 ⊢ {⟨x, y⟩ ∣ ∃zφ} = {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ ∃zφ)}
10	8, 9	eqtr4i 2060	. 2 ⊢ {w ∣ ∃z∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {⟨x, y⟩ ∣ ∃zφ}
11	2, 3, 10	3eqtri 2061	1 ⊢ dom {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨x, y⟩ ∣ ∃zφ}

Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378  {cab 2023  ⟨cop 3370  {copab 3808  dom cdm 4288  {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-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  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-br 3756  df-opab 3810  df-dm 4298  df-oprab 5459

This theorem is referenced by:  dmoprabss  5528  reldmoprab  5531  fnoprabg  5544  dmaddpq  6363  dmmulpq  6364