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Theorem cnvoprab 6210
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 1657 . . . . . 6 (∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
2 nfv 1521 . . . . . . . . . . 11 𝑥 𝑤 = ⟨𝑎, 𝑧
3 cnvoprab.x . . . . . . . . . . 11 𝑥𝜓
42, 3nfan 1558 . . . . . . . . . 10 𝑥(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
54nfex 1630 . . . . . . . . 9 𝑥𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
6 nfv 1521 . . . . . . . . . . . 12 𝑦 𝑤 = ⟨𝑎, 𝑧
7 cnvoprab.y . . . . . . . . . . . 12 𝑦𝜓
86, 7nfan 1558 . . . . . . . . . . 11 𝑦(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
98nfex 1630 . . . . . . . . . 10 𝑦𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
10 vex 2733 . . . . . . . . . . . 12 𝑥 ∈ V
11 vex 2733 . . . . . . . . . . . 12 𝑦 ∈ V
1210, 11opex 4212 . . . . . . . . . . 11 𝑥, 𝑦⟩ ∈ V
13 opeq1 3763 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑥, 𝑦⟩ → ⟨𝑎, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
1413eqeq2d 2182 . . . . . . . . . . . 12 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑤 = ⟨𝑎, 𝑧⟩ ↔ 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
15 cnvoprab.1 . . . . . . . . . . . 12 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))
1614, 15anbi12d 470 . . . . . . . . . . 11 (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
1712, 16spcev 2825 . . . . . . . . . 10 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
189, 17exlimi 1587 . . . . . . . . 9 (∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
195, 18exlimi 1587 . . . . . . . 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 2733 . . . . . . . . . . . 12 𝑎 ∈ V
23 1stexg 6143 . . . . . . . . . . . 12 (𝑎 ∈ V → (1st𝑎) ∈ V)
2422, 23ax-mp 5 . . . . . . . . . . 11 (1st𝑎) ∈ V
25 2ndexg 6144 . . . . . . . . . . . 12 (𝑎 ∈ V → (2nd𝑎) ∈ V)
2622, 25ax-mp 5 . . . . . . . . . . 11 (2nd𝑎) ∈ V
27 eqcom 2172 . . . . . . . . . . . . . . 15 ((1st𝑎) = 𝑥𝑥 = (1st𝑎))
28 eqcom 2172 . . . . . . . . . . . . . . 15 ((2nd𝑎) = 𝑦𝑦 = (2nd𝑎))
2927, 28anbi12i 457 . . . . . . . . . . . . . 14 (((1st𝑎) = 𝑥 ∧ (2nd𝑎) = 𝑦) ↔ (𝑥 = (1st𝑎) ∧ 𝑦 = (2nd𝑎)))
30 eqopi 6148 . . . . . . . . . . . . . 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 6209 . . . . . . . . . . 11 ((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ V ∧ (2nd𝑎) ∈ V)) → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3524, 26, 34mpanr12 437 . . . . . . . . . 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 1591 . . . . . . . 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 1598 . . . . . 6 (∃𝑧𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
40 exrot3 1683 . . . . . 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 2286 . . . 4 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
43 df-oprab 5854 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
44 df-opab 4049 . . . 4 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
4542, 43, 443eqtr4ri 2202 . . 3 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4645cnveqi 4784 . 2 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
47 cnvopab 5010 . 2 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
4846, 47eqtr3i 2193 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wnf 1453  wex 1485  wcel 2141  {cab 2156  Vcvv 2730  cop 3584  {copab 4047   × cxp 4607  ccnv 4608  cfv 5196  {coprab 5851  1st c1st 6114  2nd c2nd 6115
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fo 5202  df-fv 5204  df-oprab 5854  df-1st 6116  df-2nd 6117
This theorem is referenced by:  f1od2  6211
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