Theorem eloprabi 6087

Description: A consequence of membership in an operation class abstraction, using ordered pair extractors. (Contributed by NM, 6-Nov-2006.) (Revised by David Abernethy, 19-Jun-2012.)

Hypotheses

Ref	Expression
eloprabi.1	⊢ (𝑥 = (1st ‘(1st ‘𝐴)) → (𝜑 ↔ 𝜓))
eloprabi.2	⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → (𝜓 ↔ 𝜒))
eloprabi.3	⊢ (𝑧 = (2nd ‘𝐴) → (𝜒 ↔ 𝜃))

Assertion

Ref	Expression
eloprabi	⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜃,𝑥,𝑦,𝑧

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

Proof of Theorem eloprabi

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2144	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
2	1	anbi1d 460	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3	2	3exbidv 1841	. . . 4 ⊢ (𝑤 = 𝐴 → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
4		df-oprab 5771	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
5	3, 4	elab2g 2826	. . 3 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
6	5	ibi 175	. 2 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
7		vex 2684	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
8		vex 2684	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
9	7, 8	opex 4146	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
10		vex 2684	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
11	9, 10	op1std 6039	. . . . . . . . . 10 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘𝐴) = ⟨𝑥, 𝑦⟩)
12	11	fveq2d 5418	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘(1st ‘𝐴)) = (1st ‘⟨𝑥, 𝑦⟩))
13	7, 8	op1st 6037	. . . . . . . . 9 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
14	12, 13	syl6req 2187	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑥 = (1st ‘(1st ‘𝐴)))
15		eloprabi.1	. . . . . . . 8 ⊢ (𝑥 = (1st ‘(1st ‘𝐴)) → (𝜑 ↔ 𝜓))
16	14, 15	syl 14	. . . . . . 7 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 ↔ 𝜓))
17	11	fveq2d 5418	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘(1st ‘𝐴)) = (2nd ‘⟨𝑥, 𝑦⟩))
18	7, 8	op2nd 6038	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
19	17, 18	syl6req 2187	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑦 = (2nd ‘(1st ‘𝐴)))
20		eloprabi.2	. . . . . . . 8 ⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → (𝜓 ↔ 𝜒))
21	19, 20	syl 14	. . . . . . 7 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜓 ↔ 𝜒))
22	9, 10	op2ndd 6040	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘𝐴) = 𝑧)
23	22	eqcomd 2143	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑧 = (2nd ‘𝐴))
24		eloprabi.3	. . . . . . . 8 ⊢ (𝑧 = (2nd ‘𝐴) → (𝜒 ↔ 𝜃))
25	23, 24	syl 14	. . . . . . 7 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜒 ↔ 𝜃))
26	16, 21, 25	3bitrd 213	. . . . . 6 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 ↔ 𝜃))
27	26	biimpa 294	. . . . 5 ⊢ ((𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
28	27	exlimiv 1577	. . . 4 ⊢ (∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
29	28	exlimiv 1577	. . 3 ⊢ (∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
30	29	exlimiv 1577	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
31	6, 30	syl 14	1 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ⟨cop 3525  ‘cfv 5118  {coprab 5768  1^st c1st 6029  2^nd c2nd 6030

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-oprab 5771  df-1st 6031  df-2nd 6032

This theorem is referenced by: (None)