Theorem eloprabi 6422

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 2245	. . . . . 6 ⊢ (𝑤 = 𝐴 → (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ↔ 𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
2	1	anbi1d 469	. . . . 5 ⊢ (𝑤 = 𝐴 → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3	2	3exbidv 1922	. . . 4 ⊢ (𝑤 = 𝐴 → (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
4		df-oprab 6079	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
5	3, 4	elab2g 2973	. . 3 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
6	5	ibi 176	. 2 ⊢ (𝐴 ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} → ∃𝑥∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
7		vex 2824	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
8		vex 2824	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
9	7, 8	opex 4364	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
10		vex 2824	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
11	9, 10	op1std 6372	. . . . . . . . . 10 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘𝐴) = ⟨𝑥, 𝑦⟩)
12	11	fveq2d 5694	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (1st ‘(1st ‘𝐴)) = (1st ‘⟨𝑥, 𝑦⟩))
13	7, 8	op1st 6370	. . . . . . . . 9 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
14	12, 13	eqtr2di 2288	. . . . . . . 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 5694	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘(1st ‘𝐴)) = (2nd ‘⟨𝑥, 𝑦⟩))
18	7, 8	op2nd 6371	. . . . . . . . 9 ⊢ (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
19	17, 18	eqtr2di 2288	. . . . . . . 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 6373	. . . . . . . . 9 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (2nd ‘𝐴) = 𝑧)
23	22	eqcomd 2244	. . . . . . . 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 214	. . . . . 6 ⊢ (𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ → (𝜑 ↔ 𝜃))
27	26	biimpa 296	. . . . 5 ⊢ ((𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
28	27	exlimiv 1651	. . . 4 ⊢ (∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
29	28	exlimiv 1651	. . 3 ⊢ (∃𝑦∃𝑧(𝐴 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → 𝜃)
30	29	exlimiv 1651	. 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 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ⟨cop 3708  ‘cfv 5372  {coprab 6076  1^st c1st 6362  2^nd c2nd 6363

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fv 5380  df-oprab 6079  df-1st 6364  df-2nd 6365

This theorem is referenced by: (None)