Theorem cnvoprabOLD 30482

Description: The converse of a class abstraction of nested ordered pairs. Obsolete version of cnvoprab 7740 as of 16-Oct-2022, which has nonfreeness hypotheses instead of disjoint variable conditions. (Contributed by Thierry Arnoux, 17-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
cnvoprabOLD.x	⊢ Ⅎ𝑥𝜓
cnvoprabOLD.y	⊢ Ⅎ𝑦𝜓
cnvoprabOLD.1	⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜑))
cnvoprabOLD.2	⊢ (𝜓 → 𝑎 ∈ (V × V))

Assertion

Ref	Expression
cnvoprabOLD	⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Distinct variable groups:   𝑥,𝑎,𝑦,𝑧   𝜑,𝑎

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

Proof of Theorem cnvoprabOLD

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		excom 2166	. . . . . 6 ⊢ (∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
2		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 = ⟨𝑎, 𝑧⟩
3		cnvoprabOLD.x	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
5	4	nfex 2332	. . . . . . . . 9 ⊢ Ⅎ𝑥∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
6		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑤 = ⟨𝑎, 𝑧⟩
7		cnvoprabOLD.y	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦𝜓
8	6, 7	nfan 1900	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
9	8	nfex 2332	. . . . . . . . . 10 ⊢ Ⅎ𝑦∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
10		opex 5321	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
11		opeq1 4763	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ⟨𝑎, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
12	11	eqeq2d 2809	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑤 = ⟨𝑎, 𝑧⟩ ↔ 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
13		cnvoprabOLD.1	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜑))
14	12, 13	anbi12d 633	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
15	10, 14	spcev 3555	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
16	9, 15	exlimi 2215	. . . . . . . . 9 ⊢ (∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
17	5, 16	exlimi 2215	. . . . . . . 8 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
18		cnvoprabOLD.2	. . . . . . . . . . 11 ⊢ (𝜓 → 𝑎 ∈ (V × V))
19	18	adantl 485	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → 𝑎 ∈ (V × V))
20		fvex 6658	. . . . . . . . . . 11 ⊢ (1st ‘𝑎) ∈ V
21		fvex 6658	. . . . . . . . . . 11 ⊢ (2nd ‘𝑎) ∈ V
22		eqcom 2805	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑎) = 𝑥 ↔ 𝑥 = (1st ‘𝑎))
23		eqcom 2805	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑎) = 𝑦 ↔ 𝑦 = (2nd ‘𝑎))
24	22, 23	anbi12i 629	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑎) = 𝑥 ∧ (2nd ‘𝑎) = 𝑦) ↔ (𝑥 = (1st ‘𝑎) ∧ 𝑦 = (2nd ‘𝑎)))
25		eqopi 7707	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (V × V) ∧ ((1st ‘𝑎) = 𝑥 ∧ (2nd ‘𝑎) = 𝑦)) → 𝑎 = ⟨𝑥, 𝑦⟩)
26	24, 25	sylan2br 597	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑎) ∧ 𝑦 = (2nd ‘𝑎))) → 𝑎 = ⟨𝑥, 𝑦⟩)
27	14	bicomd 226	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)))
28	26, 27	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑎) ∧ 𝑦 = (2nd ‘𝑎))) → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)))
29	4, 8, 28	spc2ed 3550	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (V × V) ∧ ((1st ‘𝑎) ∈ V ∧ (2nd ‘𝑎) ∈ V)) → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
30	20, 21, 29	mpanr12 704	. . . . . . . . . 10 ⊢ (𝑎 ∈ (V × V) → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
31	19, 30	mpcom 38	. . . . . . . . 9 ⊢ ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
32	31	exlimiv 1931	. . . . . . . 8 ⊢ (∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
33	17, 32	impbii 212	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
34	33	exbii 1849	. . . . . 6 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
35		exrot3 2169	. . . . . 6 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
36	1, 34, 35	3bitr2ri 303	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
37	36	abbii 2863	. . . 4 ⊢ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
38		df-oprab 7139	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
39		df-opab 5093	. . . 4 ⊢ {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
40	37, 38, 39	3eqtr4ri 2832	. . 3 ⊢ {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
41	40	cnveqi 5709	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ 𝜓} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
42		cnvopab 5964	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
43	41, 42	eqtr3i 2823	1 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2111  {cab 2776  Vcvv 3441  ⟨cop 4531  {copab 5092   × cxp 5517  ^◡ccnv 5518  ‘cfv 6324  {coprab 7136  1^st c1st 7669  2^nd c2nd 7670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-oprab 7139  df-1st 7671  df-2nd 7672

This theorem is referenced by: (None)