Theorem cnvoprabOLD 30450

Description: The converse of a class abstraction of nested ordered pairs. Obsolete version of cnvoprab 7752 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 2165	. . . . . 6 ⊢ (∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
2		nfv 1911	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 = ⟨𝑎, 𝑧⟩
3		cnvoprabOLD.x	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfan 1896	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
5	4	nfex 2339	. . . . . . . . 9 ⊢ Ⅎ𝑥∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
6		nfv 1911	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑤 = ⟨𝑎, 𝑧⟩
7		cnvoprabOLD.y	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦𝜓
8	6, 7	nfan 1896	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
9	8	nfex 2339	. . . . . . . . . 10 ⊢ Ⅎ𝑦∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
10		opex 5348	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
11		opeq1 4796	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ⟨𝑎, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
12	11	eqeq2d 2832	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑤 = ⟨𝑎, 𝑧⟩ ↔ 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
13		cnvoprabOLD.1	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜑))
14	12, 13	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
15	10, 14	spcev 3606	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
16	9, 15	exlimi 2213	. . . . . . . . 9 ⊢ (∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
17	5, 16	exlimi 2213	. . . . . . . 8 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
18		cnvoprabOLD.2	. . . . . . . . . . 11 ⊢ (𝜓 → 𝑎 ∈ (V × V))
19	18	adantl 484	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → 𝑎 ∈ (V × V))
20		fvex 6677	. . . . . . . . . . 11 ⊢ (1st ‘𝑎) ∈ V
21		fvex 6677	. . . . . . . . . . 11 ⊢ (2nd ‘𝑎) ∈ V
22		eqcom 2828	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑎) = 𝑥 ↔ 𝑥 = (1st ‘𝑎))
23		eqcom 2828	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑎) = 𝑦 ↔ 𝑦 = (2nd ‘𝑎))
24	22, 23	anbi12i 628	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑎) = 𝑥 ∧ (2nd ‘𝑎) = 𝑦) ↔ (𝑥 = (1st ‘𝑎) ∧ 𝑦 = (2nd ‘𝑎)))
25		eqopi 7719	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (V × V) ∧ ((1st ‘𝑎) = 𝑥 ∧ (2nd ‘𝑎) = 𝑦)) → 𝑎 = ⟨𝑥, 𝑦⟩)
26	24, 25	sylan2br 596	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑎) ∧ 𝑦 = (2nd ‘𝑎))) → 𝑎 = ⟨𝑥, 𝑦⟩)
27	14	bicomd 225	. . . . . . . . . . . . 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 3601	. . . . . . . . . . 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 703	. . . . . . . . . 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 1927	. . . . . . . 8 ⊢ (∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
33	17, 32	impbii 211	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
34	33	exbii 1844	. . . . . 6 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
35		exrot3 2168	. . . . . 6 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
36	1, 34, 35	3bitr2ri 302	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
37	36	abbii 2886	. . . 4 ⊢ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
38		df-oprab 7154	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
39		df-opab 5121	. . . 4 ⊢ {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
40	37, 38, 39	3eqtr4ri 2855	. . 3 ⊢ {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
41	40	cnveqi 5739	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ 𝜓} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
42		cnvopab 5991	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
43	41, 42	eqtr3i 2846	1 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533  ∃wex 1776  Ⅎwnf 1780   ∈ wcel 2110  {cab 2799  Vcvv 3494  ⟨cop 4566  {copab 5120   × cxp 5547  ^◡ccnv 5548  ‘cfv 6349  {coprab 7151  1^st c1st 7681  2^nd c2nd 7682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-iota 6308  df-fun 6351  df-fv 6357  df-oprab 7154  df-1st 7683  df-2nd 7684

This theorem is referenced by: (None)