Theorem eloprabi 8060

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 2773	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
2	1	anbi1d 642	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3	2	3exbidv 1952	. . . 4 ⊢ (𝑤 = 𝐴 → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
4		df-oprab 7415	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
5	3, 4	elab2g 3648	. . 3 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
6	5	ibi 270	. 2 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
7		opex 5446	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
8		vex 3467	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
9	7, 8	op1std 7996	. . . . . . . . . 10 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘𝐴) = ⟨𝑥, 𝑦⟩)
10	9	fveq2d 6886	. . . . . . . . 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 7994	. . . . . . . . 9 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
14	10, 13	eqtr2di 2821	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑥 = (1st ‘(1st ‘𝐴)))
15		eloprabi.1	. . . . . . . 8 ⊢ (𝑥 = (1st ‘(1st ‘𝐴)) → (𝜑 ↔ 𝜓))
16	14, 15	syl 18	. . . . . . 7 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 ↔ 𝜓))
17	9	fveq2d 6886	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘(1st ‘𝐴)) = (2nd ‘⟨𝑥, 𝑦⟩))
18	11, 12	op2nd 7995	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
19	17, 18	eqtr2di 2821	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑦 = (2nd ‘(1st ‘𝐴)))
20		eloprabi.2	. . . . . . . 8 ⊢ (𝑦 = (2nd ‘(1st ‘𝐴)) → (𝜓 ↔ 𝜒))
21	19, 20	syl 18	. . . . . . 7 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜓 ↔ 𝜒))
22	7, 8	op2ndd 7997	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘𝐴) = 𝑧)
23	22	eqcomd 2775	. . . . . . . 8 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → 𝑧 = (2nd ‘𝐴))
24		eloprabi.3	. . . . . . . 8 ⊢ (𝑧 = (2nd ‘𝐴) → (𝜒 ↔ 𝜃))
25	23, 24	syl 18	. . . . . . 7 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜒 ↔ 𝜃))
26	16, 21, 25	3bitrd 308	. . . . . 6 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 ↔ 𝜃))
27	26	biimpa 481	. . . . 5 ⊢ ((𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
28	27	exlimiv 1957	. . . 4 ⊢ (∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
29	28	exlimiv 1957	. . 3 ⊢ (∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
30	29	exlimiv 1957	. 2 ⊢ (∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
31	6, 30	syl 18	1 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → 𝜃)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ⟨cop 4600  ‘cfv 6537  {coprab 7412  1^st c1st 7984  2^nd c2nd 7985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fv 6545  df-oprab 7415  df-1st 7986  df-2nd 7987

This theorem is referenced by:  sectpropdlem  49733  invpropdlem  49735  isopropdlem  49737