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Theorem cnvoprab 29364
Description: The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)
Hypotheses
Ref Expression
cnvoprab.x 𝑥𝜓
cnvoprab.y 𝑦𝜓
cnvoprab.1 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))
cnvoprab.2 (𝜓𝑎 ∈ (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.
StepHypRef Expression
1 excom 2039 . . . . . 6 (∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
2 nfv 1840 . . . . . . . . . . 11 𝑥 𝑤 = ⟨𝑎, 𝑧
3 cnvoprab.x . . . . . . . . . . 11 𝑥𝜓
42, 3nfan 1825 . . . . . . . . . 10 𝑥(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
54nfex 2151 . . . . . . . . 9 𝑥𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
6 nfv 1840 . . . . . . . . . . . 12 𝑦 𝑤 = ⟨𝑎, 𝑧
7 cnvoprab.y . . . . . . . . . . . 12 𝑦𝜓
86, 7nfan 1825 . . . . . . . . . . 11 𝑦(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
98nfex 2151 . . . . . . . . . 10 𝑦𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
10 opex 4898 . . . . . . . . . . 11 𝑥, 𝑦⟩ ∈ V
11 opeq1 4375 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑥, 𝑦⟩ → ⟨𝑎, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
1211eqeq2d 2631 . . . . . . . . . . . 12 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑤 = ⟨𝑎, 𝑧⟩ ↔ 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
13 cnvoprab.1 . . . . . . . . . . . 12 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))
1412, 13anbi12d 746 . . . . . . . . . . 11 (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
1510, 14spcev 3289 . . . . . . . . . 10 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
169, 15exlimi 2084 . . . . . . . . 9 (∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
175, 16exlimi 2084 . . . . . . . 8 (∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
18 cnvoprab.2 . . . . . . . . . . 11 (𝜓𝑎 ∈ (V × V))
1918adantl 482 . . . . . . . . . 10 ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → 𝑎 ∈ (V × V))
20 fvex 6163 . . . . . . . . . . 11 (1st𝑎) ∈ V
21 fvex 6163 . . . . . . . . . . 11 (2nd𝑎) ∈ V
22 eqcom 2628 . . . . . . . . . . . . . . 15 ((1st𝑎) = 𝑥𝑥 = (1st𝑎))
23 eqcom 2628 . . . . . . . . . . . . . . 15 ((2nd𝑎) = 𝑦𝑦 = (2nd𝑎))
2422, 23anbi12i 732 . . . . . . . . . . . . . 14 (((1st𝑎) = 𝑥 ∧ (2nd𝑎) = 𝑦) ↔ (𝑥 = (1st𝑎) ∧ 𝑦 = (2nd𝑎)))
25 eqopi 7154 . . . . . . . . . . . . . 14 ((𝑎 ∈ (V × V) ∧ ((1st𝑎) = 𝑥 ∧ (2nd𝑎) = 𝑦)) → 𝑎 = ⟨𝑥, 𝑦⟩)
2624, 25sylan2br 493 . . . . . . . . . . . . 13 ((𝑎 ∈ (V × V) ∧ (𝑥 = (1st𝑎) ∧ 𝑦 = (2nd𝑎))) → 𝑎 = ⟨𝑥, 𝑦⟩)
2714bicomd 213 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)))
2826, 27syl 17 . . . . . . . . . . . 12 ((𝑎 ∈ (V × V) ∧ (𝑥 = (1st𝑎) ∧ 𝑦 = (2nd𝑎))) → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)))
294, 8, 28spc2ed 29182 . . . . . . . . . . 11 ((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ V ∧ (2nd𝑎) ∈ V)) → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3020, 21, 29mpanr12 720 . . . . . . . . . 10 (𝑎 ∈ (V × V) → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3119, 30mpcom 38 . . . . . . . . 9 ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
3231exlimiv 1855 . . . . . . . 8 (∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
3317, 32impbii 199 . . . . . . 7 (∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
3433exbii 1771 . . . . . 6 (∃𝑧𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
35 exrot3 2042 . . . . . 6 (∃𝑧𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
361, 34, 353bitr2ri 289 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
3736abbii 2736 . . . 4 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
38 df-oprab 6614 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
39 df-opab 4679 . . . 4 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
4037, 38, 393eqtr4ri 2654 . . 3 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4140cnveqi 5262 . 2 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
42 cnvopab 5497 . 2 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
4341, 42eqtr3i 2645 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wex 1701  wnf 1705  wcel 1987  {cab 2607  Vcvv 3189  cop 4159  {copab 4677   × cxp 5077  ccnv 5078  cfv 5852  {coprab 6611  1st c1st 7118  2nd c2nd 7119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fv 5860  df-oprab 6614  df-1st 7120  df-2nd 7121
This theorem is referenced by:  f1od2  29365
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