Theorem cnvoprab 6301

Description: The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)

Hypotheses

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

Assertion

Ref	Expression
cnvoprab	⊢ ◡{⟨⟨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 cnvoprab

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		excom 1678	. . . . . 6 ⊢ (∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
2		nfv 1542	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 = ⟨𝑎, 𝑧⟩
3		cnvoprab.x	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝜓
4	2, 3	nfan 1579	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
5	4	nfex 1651	. . . . . . . . 9 ⊢ Ⅎ𝑥∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
6		nfv 1542	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦 𝑤 = ⟨𝑎, 𝑧⟩
7		cnvoprab.y	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦𝜓
8	6, 7	nfan 1579	. . . . . . . . . . 11 ⊢ Ⅎ𝑦(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
9	8	nfex 1651	. . . . . . . . . 10 ⊢ Ⅎ𝑦∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
10		vex 2766	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
11		vex 2766	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
12	10, 11	opex 4263	. . . . . . . . . . 11 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
13		opeq1 3809	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ⟨𝑎, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
14	13	eqeq2d 2208	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑤 = ⟨𝑎, 𝑧⟩ ↔ 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
15		cnvoprab.1	. . . . . . . . . . . 12 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓 ↔ 𝜑))
16	14, 15	anbi12d 473	. . . . . . . . . . 11 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
17	12, 16	spcev 2859	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
18	9, 17	exlimi 1608	. . . . . . . . 9 ⊢ (∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
19	5, 18	exlimi 1608	. . . . . . . 8 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
20		cnvoprab.2	. . . . . . . . . . 11 ⊢ (𝜓 → 𝑎 ∈ (V × V))
21	20	adantl 277	. . . . . . . . . 10 ⊢ ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → 𝑎 ∈ (V × V))
22		vex 2766	. . . . . . . . . . . 12 ⊢ 𝑎 ∈ V
23		1stexg 6234	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ V → (1st ‘𝑎) ∈ V)
24	22, 23	ax-mp 5	. . . . . . . . . . 11 ⊢ (1st ‘𝑎) ∈ V
25		2ndexg 6235	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ V → (2nd ‘𝑎) ∈ V)
26	22, 25	ax-mp 5	. . . . . . . . . . 11 ⊢ (2nd ‘𝑎) ∈ V
27		eqcom 2198	. . . . . . . . . . . . . . 15 ⊢ ((1st ‘𝑎) = 𝑥 ↔ 𝑥 = (1st ‘𝑎))
28		eqcom 2198	. . . . . . . . . . . . . . 15 ⊢ ((2nd ‘𝑎) = 𝑦 ↔ 𝑦 = (2nd ‘𝑎))
29	27, 28	anbi12i 460	. . . . . . . . . . . . . 14 ⊢ (((1st ‘𝑎) = 𝑥 ∧ (2nd ‘𝑎) = 𝑦) ↔ (𝑥 = (1st ‘𝑎) ∧ 𝑦 = (2nd ‘𝑎)))
30		eqopi 6239	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ (V × V) ∧ ((1st ‘𝑎) = 𝑥 ∧ (2nd ‘𝑎) = 𝑦)) → 𝑎 = ⟨𝑥, 𝑦⟩)
31	29, 30	sylan2br 288	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑎) ∧ 𝑦 = (2nd ‘𝑎))) → 𝑎 = ⟨𝑥, 𝑦⟩)
32	16	bicomd 141	. . . . . . . . . . . . 13 ⊢ (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)))
33	31, 32	syl 14	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ (V × V) ∧ (𝑥 = (1st ‘𝑎) ∧ 𝑦 = (2nd ‘𝑎))) → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)))
34	4, 8, 33	spc2ed 6300	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ (V × V) ∧ ((1st ‘𝑎) ∈ V ∧ (2nd ‘𝑎) ∈ V)) → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
35	24, 26, 34	mpanr12 439	. . . . . . . . . 10 ⊢ (𝑎 ∈ (V × V) → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
36	21, 35	mpcom 36	. . . . . . . . 9 ⊢ ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
37	36	exlimiv 1612	. . . . . . . 8 ⊢ (∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
38	19, 37	impbii 126	. . . . . . 7 ⊢ (∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
39	38	exbii 1619	. . . . . 6 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
40		exrot3 1704	. . . . . 6 ⊢ (∃𝑧∃𝑥∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
41	1, 39, 40	3bitr2ri 209	. . . . 5 ⊢ (∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
42	41	abbii 2312	. . . 4 ⊢ {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
43		df-oprab 5929	. . . 4 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥∃𝑦∃𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
44		df-opab 4096	. . . 4 ⊢ {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑎∃𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
45	42, 43, 44	3eqtr4ri 2228	. . 3 ⊢ {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
46	45	cnveqi 4842	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ 𝜓} = ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
47		cnvopab 5072	. 2 ⊢ ◡{⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
48	46, 47	eqtr3i 2219	1 ⊢ ◡{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  Ⅎwnf 1474  ∃wex 1506   ∈ wcel 2167  {cab 2182  Vcvv 2763  ⟨cop 3626  {copab 4094   × cxp 4662  ^◡ccnv 4663  ‘cfv 5259  {coprab 5926  1^st c1st 6205  2^nd c2nd 6206

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fo 5265  df-fv 5267  df-oprab 5929  df-1st 6207  df-2nd 6208

This theorem is referenced by:  f1od2  6302