Theorem dfoprab3 6290

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 6289	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓)}
2		vex 2776	. . . . . 6 ⊢ 𝑤 ∈ V
3		1stexg 6266	. . . . . 6 ⊢ (𝑤 ∈ V → (1st ‘𝑤) ∈ V)
4	2, 3	ax-mp 5	. . . . 5 ⊢ (1st ‘𝑤) ∈ V
5		2ndexg 6267	. . . . . 6 ⊢ (𝑤 ∈ V → (2nd ‘𝑤) ∈ V)
6	2, 5	ax-mp 5	. . . . 5 ⊢ (2nd ‘𝑤) ∈ V
7		eqcom 2208	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑤) ↔ (1st ‘𝑤) = 𝑥)
8		eqcom 2208	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑤) ↔ (2nd ‘𝑤) = 𝑦)
9	7, 8	anbi12i 460	. . . . . . . . 9 ⊢ ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) ↔ ((1st ‘𝑤) = 𝑥 ∧ (2nd ‘𝑤) = 𝑦))
10		eqopi 6271	. . . . . . . . 9 ⊢ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) = 𝑥 ∧ (2nd ‘𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
11	9, 10	sylan2b 287	. . . . . . . 8 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → 𝑤 = ⟨𝑥, 𝑦⟩)
12		dfoprab3.1	. . . . . . . 8 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
13	11, 12	syl 14	. . . . . . 7 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝜑 ↔ 𝜓))
14	13	bicomd 141	. . . . . 6 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝜓 ↔ 𝜑))
15	14	ex 115	. . . . 5 ⊢ (𝑤 ∈ (V × V) → ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (𝜓 ↔ 𝜑)))
16	4, 6, 15	sbc2iedv 3075	. . . 4 ⊢ (𝑤 ∈ (V × V) → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓 ↔ 𝜑))
17	16	pm5.32i 454	. . 3 ⊢ ((𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑))
18	17	opabbii 4119	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)}
19	1, 18	eqtr2i 2228	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2177  Vcvv 2773  [wsbc 3002  ⟨cop 3641  {copab 4112   × cxp 4681  ‘cfv 5280  {coprab 5958  1^st c1st 6237  2^nd c2nd 6238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fo 5286  df-fv 5288  df-oprab 5961  df-1st 6239  df-2nd 6240

This theorem is referenced by:  dfoprab4  6291  df1st2  6318  df2nd2  6319