Theorem eloprabi 8070

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 2738	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
2	1	anbi1d 631	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3	2	3exbidv 1924	. . . 4 ⊢ (𝑤 = 𝐴 → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
4		df-oprab 7417	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
5	3, 4	elab2g 3663	. . 3 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
6	5	ibi 267	. 2 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
7		opex 5449	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
8		vex 3467	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
9	7, 8	op1std 8006	. . . . . . . . . 10 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘𝐴) = ⟨𝑥, 𝑦⟩)
10	9	fveq2d 6890	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘(1st ‘𝐴)) = (1st ‘⟨𝑥, 𝑦⟩))
11		vex 3467	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
12		vex 3467	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
13	11, 12	op1st 8004	. . . . . . . . 9 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
14	10, 13	eqtr2di 2786	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑥 = (1st ‘(1st ‘𝐴)))
15		eloprabi.1	. . . . . . . 8 ⊢ (𝑥 = (1st ‘(1st ‘𝐴)) → (𝜑 ↔ 𝜓))
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 ↔ 𝜓))
17	9	fveq2d 6890	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘(1st ‘𝐴)) = (2nd ‘⟨𝑥, 𝑦⟩))
18	11, 12	op2nd 8005	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
19	17, 18	eqtr2di 2786	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑦 = (2nd ‘(1st ‘𝐴)))
20		eloprabi.2	. . . . . . . 8 ⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → (𝜓 ↔ 𝜒))
21	19, 20	syl 17	. . . . . . 7 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜓 ↔ 𝜒))
22	7, 8	op2ndd 8007	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘𝐴) = 𝑧)
23	22	eqcomd 2740	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑧 = (2nd ‘𝐴))
24		eloprabi.3	. . . . . . . 8 ⊢ (𝑧 = (2nd ‘𝐴) → (𝜒 ↔ 𝜃))
25	23, 24	syl 17	. . . . . . 7 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜒 ↔ 𝜃))
26	16, 21, 25	3bitrd 305	. . . . . 6 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 ↔ 𝜃))
27	26	biimpa 476	. . . . 5 ⊢ ((𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
28	27	exlimiv 1929	. . . 4 ⊢ (∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
29	28	exlimiv 1929	. . 3 ⊢ (∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
30	29	exlimiv 1929	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
31	6, 30	syl 17	1 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ⟨cop 4612  ‘cfv 6541  {coprab 7414  1^st c1st 7994  2^nd c2nd 7995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-iota 6494  df-fun 6543  df-fv 6549  df-oprab 7417  df-1st 7996  df-2nd 7997

This theorem is referenced by: (None)