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Theorem cnvoprab 6099
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 1627 . . . . . 6 (∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
2 nfv 1493 . . . . . . . . . . 11 𝑥 𝑤 = ⟨𝑎, 𝑧
3 cnvoprab.x . . . . . . . . . . 11 𝑥𝜓
42, 3nfan 1529 . . . . . . . . . 10 𝑥(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
54nfex 1601 . . . . . . . . 9 𝑥𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
6 nfv 1493 . . . . . . . . . . . 12 𝑦 𝑤 = ⟨𝑎, 𝑧
7 cnvoprab.y . . . . . . . . . . . 12 𝑦𝜓
86, 7nfan 1529 . . . . . . . . . . 11 𝑦(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
98nfex 1601 . . . . . . . . . 10 𝑦𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
10 vex 2663 . . . . . . . . . . . 12 𝑥 ∈ V
11 vex 2663 . . . . . . . . . . . 12 𝑦 ∈ V
1210, 11opex 4121 . . . . . . . . . . 11 𝑥, 𝑦⟩ ∈ V
13 opeq1 3675 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑥, 𝑦⟩ → ⟨𝑎, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
1413eqeq2d 2129 . . . . . . . . . . . 12 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑤 = ⟨𝑎, 𝑧⟩ ↔ 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
15 cnvoprab.1 . . . . . . . . . . . 12 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))
1614, 15anbi12d 464 . . . . . . . . . . 11 (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
1712, 16spcev 2754 . . . . . . . . . 10 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
189, 17exlimi 1558 . . . . . . . . 9 (∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
195, 18exlimi 1558 . . . . . . . 8 (∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
20 cnvoprab.2 . . . . . . . . . . 11 (𝜓𝑎 ∈ (V × V))
2120adantl 275 . . . . . . . . . 10 ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → 𝑎 ∈ (V × V))
22 vex 2663 . . . . . . . . . . . 12 𝑎 ∈ V
23 1stexg 6033 . . . . . . . . . . . 12 (𝑎 ∈ V → (1st𝑎) ∈ V)
2422, 23ax-mp 5 . . . . . . . . . . 11 (1st𝑎) ∈ V
25 2ndexg 6034 . . . . . . . . . . . 12 (𝑎 ∈ V → (2nd𝑎) ∈ V)
2622, 25ax-mp 5 . . . . . . . . . . 11 (2nd𝑎) ∈ V
27 eqcom 2119 . . . . . . . . . . . . . . 15 ((1st𝑎) = 𝑥𝑥 = (1st𝑎))
28 eqcom 2119 . . . . . . . . . . . . . . 15 ((2nd𝑎) = 𝑦𝑦 = (2nd𝑎))
2927, 28anbi12i 455 . . . . . . . . . . . . . 14 (((1st𝑎) = 𝑥 ∧ (2nd𝑎) = 𝑦) ↔ (𝑥 = (1st𝑎) ∧ 𝑦 = (2nd𝑎)))
30 eqopi 6038 . . . . . . . . . . . . . 14 ((𝑎 ∈ (V × V) ∧ ((1st𝑎) = 𝑥 ∧ (2nd𝑎) = 𝑦)) → 𝑎 = ⟨𝑥, 𝑦⟩)
3129, 30sylan2br 286 . . . . . . . . . . . . 13 ((𝑎 ∈ (V × V) ∧ (𝑥 = (1st𝑎) ∧ 𝑦 = (2nd𝑎))) → 𝑎 = ⟨𝑥, 𝑦⟩)
3216bicomd 140 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)))
3331, 32syl 14 . . . . . . . . . . . 12 ((𝑎 ∈ (V × V) ∧ (𝑥 = (1st𝑎) ∧ 𝑦 = (2nd𝑎))) → ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)))
344, 8, 33spc2ed 6098 . . . . . . . . . . 11 ((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ V ∧ (2nd𝑎) ∈ V)) → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3524, 26, 34mpanr12 435 . . . . . . . . . 10 (𝑎 ∈ (V × V) → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3621, 35mpcom 36 . . . . . . . . 9 ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
3736exlimiv 1562 . . . . . . . 8 (∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
3819, 37impbii 125 . . . . . . 7 (∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
3938exbii 1569 . . . . . 6 (∃𝑧𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
40 exrot3 1653 . . . . . 6 (∃𝑧𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑))
411, 39, 403bitr2ri 208 . . . . 5 (∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
4241abbii 2233 . . . 4 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
43 df-oprab 5746 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
44 df-opab 3960 . . . 4 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
4542, 43, 443eqtr4ri 2149 . . 3 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4645cnveqi 4684 . 2 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
47 cnvopab 4910 . 2 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
4846, 47eqtr3i 2140 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wnf 1421  wex 1453  wcel 1465  {cab 2103  Vcvv 2660  cop 3500  {copab 3958   × cxp 4507  ccnv 4508  cfv 5093  {coprab 5743  1st c1st 6004  2nd c2nd 6005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fo 5099  df-fv 5101  df-oprab 5746  df-1st 6006  df-2nd 6007
This theorem is referenced by:  f1od2  6100
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