Theorem dfoprab3s 8032

Description: A way to define an operation class abstraction without using existential quantifiers. (Contributed by NM, 18-Aug-2006.) (Revised by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
dfoprab3s	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)}

Distinct variable groups:   𝜑,𝑤   𝑥,𝑦,𝑧,𝑤

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

Proof of Theorem dfoprab3s

Step	Hyp	Ref	Expression
1		dfoprab2 7447	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfsbc1v 3773	. . . . 5 ⊢ Ⅎ𝑥[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑
3	2	19.41 2236	. . . 4 ⊢ (∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
4		sbcopeq1a 8028	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑 ↔ 𝜑))
5	4	pm5.32i 574	. . . . . . 7 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6	5	exbii 1848	. . . . . 6 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
7		nfcv 2891	. . . . . . . 8 ⊢ Ⅎ𝑦(1st ‘𝑤)
8		nfsbc1v 3773	. . . . . . . 8 ⊢ Ⅎ𝑦[(2nd ‘𝑤) / 𝑦]𝜑
9	7, 8	nfsbcw 3775	. . . . . . 7 ⊢ Ⅎ𝑦[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑
10	9	19.41 2236	. . . . . 6 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
11	6, 10	bitr3i 277	. . . . 5 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
12	11	exbii 1848	. . . 4 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
13		elvv 5713	. . . . 5 ⊢ (𝑤 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
14	13	anbi1i 624	. . . 4 ⊢ ((𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
15	3, 12, 14	3bitr4i 303	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
16	15	opabbii 5174	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)}
17	1, 16	eqtri 2752	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)}

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3447  [wsbc 3753  ⟨cop 4595  {copab 5169   × cxp 5636  ‘cfv 6511  {coprab 7388  1^st c1st 7966  2^nd c2nd 7967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-oprab 7391  df-1st 7968  df-2nd 7969

This theorem is referenced by:  dfoprab3  8033