Theorem dfoprab3 7751

Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 16-Dec-2008.)

Hypothesis

Ref	Expression
dfoprab3.1	⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
dfoprab3	⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

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

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

Proof of Theorem dfoprab3

Step	Hyp	Ref	Expression
1		dfoprab3s 7750	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓)}
2		fvex 6682	. . . . 5 ⊢ (1st ‘𝑤) ∈ V
3		fvex 6682	. . . . 5 ⊢ (2nd ‘𝑤) ∈ V
4		eqcom 2828	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑤) ↔ (1st ‘𝑤) = 𝑥)
5		eqcom 2828	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑤) ↔ (2nd ‘𝑤) = 𝑦)
6	4, 5	anbi12i 628	. . . . . . . . 9 ⊢ ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) ↔ ((1st ‘𝑤) = 𝑥 ∧ (2nd ‘𝑤) = 𝑦))
7		eqopi 7724	. . . . . . . . 9 ⊢ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) = 𝑥 ∧ (2nd ‘𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
8	6, 7	sylan2b 595	. . . . . . . 8 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → 𝑤 = ⟨𝑥, 𝑦⟩)
9		dfoprab3.1	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
10	8, 9	syl 17	. . . . . . 7 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝜑 ↔ 𝜓))
11	10	bicomd 225	. . . . . 6 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝜓 ↔ 𝜑))
12	11	ex 415	. . . . 5 ⊢ (𝑤 ∈ (V × V) → ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (𝜓 ↔ 𝜑)))
13	2, 3, 12	sbc2iedv 3850	. . . 4 ⊢ (𝑤 ∈ (V × V) → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓 ↔ 𝜑))
14	13	pm5.32i 577	. . 3 ⊢ ((𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑))
15	14	opabbii 5132	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)}
16	1, 15	eqtr2i 2845	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494  [wsbc 3771  ⟨cop 4572  {copab 5127   × cxp 5552  ‘cfv 6354  {coprab 7156  1^st c1st 7686  2^nd c2nd 7687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fv 6362  df-oprab 7159  df-1st 7688  df-2nd 7689

This theorem is referenced by:  dfoprab4  7752  cnvoprab  7757  df1st2  7792  df2nd2  7793