Theorem dfoprab3 6159

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 6158	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓)}
2		vex 2729	. . . . . 6 ⊢ 𝑤 ∈ V
3		1stexg 6135	. . . . . 6 ⊢ (𝑤 ∈ V → (1st ‘𝑤) ∈ V)
4	2, 3	ax-mp 5	. . . . 5 ⊢ (1st ‘𝑤) ∈ V
5		2ndexg 6136	. . . . . 6 ⊢ (𝑤 ∈ V → (2nd ‘𝑤) ∈ V)
6	2, 5	ax-mp 5	. . . . 5 ⊢ (2nd ‘𝑤) ∈ V
7		eqcom 2167	. . . . . . . . . 10 ⊢ (𝑥 = (1st ‘𝑤) ↔ (1st ‘𝑤) = 𝑥)
8		eqcom 2167	. . . . . . . . . 10 ⊢ (𝑦 = (2nd ‘𝑤) ↔ (2nd ‘𝑤) = 𝑦)
9	7, 8	anbi12i 456	. . . . . . . . 9 ⊢ ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) ↔ ((1st ‘𝑤) = 𝑥 ∧ (2nd ‘𝑤) = 𝑦))
10		eqopi 6140	. . . . . . . . 9 ⊢ ((𝑤 ∈ (V × V) ∧ ((1st ‘𝑤) = 𝑥 ∧ (2nd ‘𝑤) = 𝑦)) → 𝑤 = ⟨𝑥, 𝑦⟩)
11	9, 10	sylan2b 285	. . . . . . . 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 140	. . . . . 6 ⊢ ((𝑤 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤))) → (𝜓 ↔ 𝜑))
15	14	ex 114	. . . . 5 ⊢ (𝑤 ∈ (V × V) → ((𝑥 = (1st ‘𝑤) ∧ 𝑦 = (2nd ‘𝑤)) → (𝜓 ↔ 𝜑)))
16	4, 6, 15	sbc2iedv 3023	. . . 4 ⊢ (𝑤 ∈ (V × V) → ([(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓 ↔ 𝜑))
17	16	pm5.32i 450	. . 3 ⊢ ((𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓) ↔ (𝑤 ∈ (V × V) ∧ 𝜑))
18	17	opabbii 4049	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ [(1st ‘𝑤) / 𝑥][(2nd ‘𝑤) / 𝑦]𝜓)} = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)}
19	1, 18	eqtr2i 2187	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  Vcvv 2726  [wsbc 2951  ⟨cop 3579  {copab 4042   × cxp 4602  ‘cfv 5188  {coprab 5843  1^st c1st 6106  2^nd c2nd 6107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fo 5194  df-fv 5196  df-oprab 5846  df-1st 6108  df-2nd 6109

This theorem is referenced by:  dfoprab4  6160  df1st2  6187  df2nd2  6188