Theorem dfoprab4f 6189

Description: Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
dfoprab4f.x	⊢ Ⅎ𝑥𝜑
dfoprab4f.y	⊢ Ⅎ𝑦𝜑
dfoprab4f.1	⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
dfoprab4f	⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑤,𝐴,𝑥,𝑦   𝑤,𝐵,𝑥,𝑦   𝜓,𝑤

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

Proof of Theorem dfoprab4f

Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1528	. . . . 5 ⊢ Ⅎ𝑥 𝑤 = ⟨𝑡, 𝑢⟩
2		dfoprab4f.x	. . . . . 6 ⊢ Ⅎ𝑥𝜑
3		nfs1v 1939	. . . . . 6 ⊢ Ⅎ𝑥[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
4	2, 3	nfbi 1589	. . . . 5 ⊢ Ⅎ𝑥(𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
5	1, 4	nfim 1572	. . . 4 ⊢ Ⅎ𝑥(𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
6		opeq1 3777	. . . . . 6 ⊢ (𝑥 = 𝑡 → ⟨𝑥, 𝑢⟩ = ⟨𝑡, 𝑢⟩)
7	6	eqeq2d 2189	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑤 = ⟨𝑥, 𝑢⟩ ↔ 𝑤 = ⟨𝑡, 𝑢⟩))
8		sbequ12 1771	. . . . . 6 ⊢ (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
9	8	bibi2d 232	. . . . 5 ⊢ (𝑥 = 𝑡 → ((𝜑 ↔ [𝑢 / 𝑦]𝜓) ↔ (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
10	7, 9	imbi12d 234	. . . 4 ⊢ (𝑥 = 𝑡 → ((𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓)) ↔ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))))
11		nfv 1528	. . . . . 6 ⊢ Ⅎ𝑦 𝑤 = ⟨𝑥, 𝑢⟩
12		dfoprab4f.y	. . . . . . 7 ⊢ Ⅎ𝑦𝜑
13		nfs1v 1939	. . . . . . 7 ⊢ Ⅎ𝑦[𝑢 / 𝑦]𝜓
14	12, 13	nfbi 1589	. . . . . 6 ⊢ Ⅎ𝑦(𝜑 ↔ [𝑢 / 𝑦]𝜓)
15	11, 14	nfim 1572	. . . . 5 ⊢ Ⅎ𝑦(𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
16		opeq2 3778	. . . . . . 7 ⊢ (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
17	16	eqeq2d 2189	. . . . . 6 ⊢ (𝑦 = 𝑢 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑢⟩))
18		sbequ12 1771	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (𝜓 ↔ [𝑢 / 𝑦]𝜓))
19	18	bibi2d 232	. . . . . 6 ⊢ (𝑦 = 𝑢 → ((𝜑 ↔ 𝜓) ↔ (𝜑 ↔ [𝑢 / 𝑦]𝜓)))
20	17, 19	imbi12d 234	. . . . 5 ⊢ (𝑦 = 𝑢 → ((𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓)) ↔ (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))))
21		dfoprab4f.1	. . . . 5 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑 ↔ 𝜓))
22	15, 20, 21	chvar 1757	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
23	5, 10, 22	chvar 1757	. . 3 ⊢ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
24	23	dfoprab4 6188	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
25		nfv 1528	. . 3 ⊢ Ⅎ𝑡((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)
26		nfv 1528	. . 3 ⊢ Ⅎ𝑢((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)
27		nfv 1528	. . . 4 ⊢ Ⅎ𝑥(𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵)
28	27, 3	nfan 1565	. . 3 ⊢ Ⅎ𝑥((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
29		nfv 1528	. . . 4 ⊢ Ⅎ𝑦(𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵)
30	13	nfsb 1946	. . . 4 ⊢ Ⅎ𝑦[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
31	29, 30	nfan 1565	. . 3 ⊢ Ⅎ𝑦((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
32		eleq1 2240	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝑥 ∈ 𝐴 ↔ 𝑡 ∈ 𝐴))
33		eleq1 2240	. . . . 5 ⊢ (𝑦 = 𝑢 → (𝑦 ∈ 𝐵 ↔ 𝑢 ∈ 𝐵))
34	32, 33	bi2anan9 606	. . . 4 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵)))
35	18, 8	sylan9bbr 463	. . . 4 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
36	34, 35	anbi12d 473	. . 3 ⊢ ((𝑥 = 𝑡 ∧ 𝑦 = 𝑢) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓) ↔ ((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
37	25, 26, 28, 31, 36	cbvoprab12 5944	. 2 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡 ∈ 𝐴 ∧ 𝑢 ∈ 𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
38	24, 37	eqtr4i 2201	1 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜓)}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353  Ⅎwnf 1460  [wsb 1762   ∈ wcel 2148  ⟨cop 3595  {copab 4061   × cxp 4622  {coprab 5871

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207  ax-un 4431

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3809  df-br 4002  df-opab 4063  df-mpt 4064  df-id 4291  df-xp 4630  df-rel 4631  df-cnv 4632  df-co 4633  df-dm 4634  df-rn 4635  df-iota 5175  df-fun 5215  df-fn 5216  df-f 5217  df-fo 5219  df-fv 5221  df-oprab 5874  df-1st 6136  df-2nd 6137

This theorem is referenced by: (None)