Theorem dfoprab3s 8067

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 7483	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ ∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑)}
2		nfsbc1v 3796	. . . . 5 ⊢ Ⅎ𝑥[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑
3	2	19.41 2224	. . . 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 8063	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑 ↔ 𝜑))
5	4	pm5.32i 573	. . . . . . 7 ⊢ ((𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ (𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
6	5	exbii 1843	. . . . . 6 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑))
7		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑦(1st ‘𝑤)
8		nfsbc1v 3796	. . . . . . . 8 ⊢ Ⅎ𝑦[(2nd ‘𝑤) / 𝑦]𝜑
9	7, 8	nfsbcw 3798	. . . . . . 7 ⊢ Ⅎ𝑦[(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑
10	9	19.41 2224	. . . . . 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 276	. . . . 5 ⊢ (∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
12	11	exbii 1843	. . . 4 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ ∃𝑥(∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩ ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
13		elvv 5756	. . . . 5 ⊢ (𝑤 ∈ (V × V) ↔ ∃𝑥∃𝑦 𝑤 = ⟨𝑥, 𝑦⟩)
14	13	anbi1i 622	. . . 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 302	. . 3 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜑) ↔ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑))
16	15	opabbii 5220	. 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 2754	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜑)}

Syntax hints:   ∧ wa 394   = wceq 1534  ∃wex 1774   ∈ wcel 2099  Vcvv 3462  [wsbc 3776  ⟨cop 4639  {copab 5215   × cxp 5680  ‘cfv 6554  {coprab 7425  1^st c1st 8001  2^nd c2nd 8002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6506  df-fun 6556  df-fv 6562  df-oprab 7428  df-1st 8003  df-2nd 8004

This theorem is referenced by:  dfoprab3  8068