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Theorem cnvoprabOLD 31923
Description: The converse of a class abstraction of nested ordered pairs. Obsolete version of cnvoprab 8041 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 𝑥𝜓
cnvoprabOLD.y 𝑦𝜓
cnvoprabOLD.1 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))
cnvoprabOLD.2 (𝜓𝑎 ∈ (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.
StepHypRef Expression
1 excom 2163 . . . . . 6 (∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
2 nfv 1918 . . . . . . . . . . 11 𝑥 𝑤 = ⟨𝑎, 𝑧
3 cnvoprabOLD.x . . . . . . . . . . 11 𝑥𝜓
42, 3nfan 1903 . . . . . . . . . 10 𝑥(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
54nfex 2318 . . . . . . . . 9 𝑥𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
6 nfv 1918 . . . . . . . . . . . 12 𝑦 𝑤 = ⟨𝑎, 𝑧
7 cnvoprabOLD.y . . . . . . . . . . . 12 𝑦𝜓
86, 7nfan 1903 . . . . . . . . . . 11 𝑦(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
98nfex 2318 . . . . . . . . . 10 𝑦𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
10 opex 5463 . . . . . . . . . . 11 𝑥, 𝑦⟩ ∈ V
11 opeq1 4872 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑥, 𝑦⟩ → ⟨𝑎, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
1211eqeq2d 2744 . . . . . . . . . . . 12 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑤 = ⟨𝑎, 𝑧⟩ ↔ 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
13 cnvoprabOLD.1 . . . . . . . . . . . 12 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))
1412, 13anbi12d 632 . . . . . . . . . . 11 (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
1510, 14spcev 3596 . . . . . . . . . 10 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
169, 15exlimi 2211 . . . . . . . . 9 (∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
175, 16exlimi 2211 . . . . . . . 8 (∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
18 cnvoprabOLD.2 . . . . . . . . . . 11 (𝜓𝑎 ∈ (V × V))
1918adantl 483 . . . . . . . . . 10 ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → 𝑎 ∈ (V × V))
20 fvex 6901 . . . . . . . . . . 11 (1st𝑎) ∈ V
21 fvex 6901 . . . . . . . . . . 11 (2nd𝑎) ∈ V
22 eqcom 2740 . . . . . . . . . . . . . . 15 ((1st𝑎) = 𝑥𝑥 = (1st𝑎))
23 eqcom 2740 . . . . . . . . . . . . . . 15 ((2nd𝑎) = 𝑦𝑦 = (2nd𝑎))
2422, 23anbi12i 628 . . . . . . . . . . . . . 14 (((1st𝑎) = 𝑥 ∧ (2nd𝑎) = 𝑦) ↔ (𝑥 = (1st𝑎) ∧ 𝑦 = (2nd𝑎)))
25 eqopi 8006 . . . . . . . . . . . . . 14 ((𝑎 ∈ (V × V) ∧ ((1st𝑎) = 𝑥 ∧ (2nd𝑎) = 𝑦)) → 𝑎 = ⟨𝑥, 𝑦⟩)
2624, 25sylan2br 596 . . . . . . . . . . . . 13 ((𝑎 ∈ (V × V) ∧ (𝑥 = (1st𝑎) ∧ 𝑦 = (2nd𝑎))) → 𝑎 = ⟨𝑥, 𝑦⟩)
2714bicomd 222 . . . . . . . . . . . . 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 3591 . . . . . . . . . . 11 ((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ V ∧ (2nd𝑎) ∈ V)) → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3020, 21, 29mpanr12 704 . . . . . . . . . 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 1934 . . . . . . . 8 (∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
3317, 32impbii 208 . . . . . . 7 (∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
3433exbii 1851 . . . . . 6 (∃𝑧𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
35 exrot3 2166 . . . . . 6 (∃𝑧𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
361, 34, 353bitr2ri 300 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
3736abbii 2803 . . . 4 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
38 df-oprab 7408 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
39 df-opab 5210 . . . 4 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
4037, 38, 393eqtr4ri 2772 . . 3 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4140cnveqi 5872 . 2 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
42 cnvopab 6135 . 2 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
4341, 42eqtr3i 2763 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wex 1782  wnf 1786  wcel 2107  {cab 2710  Vcvv 3475  cop 4633  {copab 5209   × cxp 5673  ccnv 5674  cfv 6540  {coprab 7405  1st c1st 7968  2nd c2nd 7969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-oprab 7408  df-1st 7970  df-2nd 7971
This theorem is referenced by: (None)
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