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Theorem cnvoprabOLD 29728
 Description: The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.) (New usage is discouraged.) (Proof modification 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 2155 . . . . . 6 (∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ ∃𝑧𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
2 nfv 1956 . . . . . . . . . . 11 𝑥 𝑤 = ⟨𝑎, 𝑧
3 cnvoprabOLD.x . . . . . . . . . . 11 𝑥𝜓
42, 3nfan 1941 . . . . . . . . . 10 𝑥(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
54nfex 2265 . . . . . . . . 9 𝑥𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
6 nfv 1956 . . . . . . . . . . . 12 𝑦 𝑤 = ⟨𝑎, 𝑧
7 cnvoprabOLD.y . . . . . . . . . . . 12 𝑦𝜓
86, 7nfan 1941 . . . . . . . . . . 11 𝑦(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
98nfex 2265 . . . . . . . . . 10 𝑦𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)
10 opex 5037 . . . . . . . . . . 11 𝑥, 𝑦⟩ ∈ V
11 opeq1 4509 . . . . . . . . . . . . 13 (𝑎 = ⟨𝑥, 𝑦⟩ → ⟨𝑎, 𝑧⟩ = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩)
1211eqeq2d 2734 . . . . . . . . . . . 12 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝑤 = ⟨𝑎, 𝑧⟩ ↔ 𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩))
13 cnvoprabOLD.1 . . . . . . . . . . . 12 (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))
1412, 13anbi12d 749 . . . . . . . . . . 11 (𝑎 = ⟨𝑥, 𝑦⟩ → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) ↔ (𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
1510, 14spcev 3404 . . . . . . . . . 10 ((𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
169, 15exlimi 2197 . . . . . . . . 9 (∃𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
175, 16exlimi 2197 . . . . . . . 8 (∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) → ∃𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
18 cnvoprabOLD.2 . . . . . . . . . . 11 (𝜓𝑎 ∈ (V × V))
1918adantl 473 . . . . . . . . . 10 ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → 𝑎 ∈ (V × V))
20 fvex 6314 . . . . . . . . . . 11 (1st𝑎) ∈ V
21 fvex 6314 . . . . . . . . . . 11 (2nd𝑎) ∈ V
22 eqcom 2731 . . . . . . . . . . . . . . 15 ((1st𝑎) = 𝑥𝑥 = (1st𝑎))
23 eqcom 2731 . . . . . . . . . . . . . . 15 ((2nd𝑎) = 𝑦𝑦 = (2nd𝑎))
2422, 23anbi12i 735 . . . . . . . . . . . . . 14 (((1st𝑎) = 𝑥 ∧ (2nd𝑎) = 𝑦) ↔ (𝑥 = (1st𝑎) ∧ 𝑦 = (2nd𝑎)))
25 eqopi 7321 . . . . . . . . . . . . . 14 ((𝑎 ∈ (V × V) ∧ ((1st𝑎) = 𝑥 ∧ (2nd𝑎) = 𝑦)) → 𝑎 = ⟨𝑥, 𝑦⟩)
2624, 25sylan2br 494 . . . . . . . . . . . . 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 29542 . . . . . . . . . . 11 ((𝑎 ∈ (V × V) ∧ ((1st𝑎) ∈ V ∧ (2nd𝑎) ∈ V)) → ((𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓) → ∃𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)))
3020, 21, 29mpanr12 723 . . . . . . . . . 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 1971 . . . . . . . 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 1887 . . . . . 6 (∃𝑧𝑥𝑦(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧𝑎(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓))
35 exrot3 2158 . . . . . 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 2841 . . . 4 {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
38 df-oprab 6769 . . . 4 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑦𝑧(𝑤 = ⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∧ 𝜑)}
39 df-opab 4821 . . . 4 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑎𝑧(𝑤 = ⟨𝑎, 𝑧⟩ ∧ 𝜓)}
4037, 38, 393eqtr4ri 2757 . . 3 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
4140cnveqi 5404 . 2 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
42 cnvopab 5643 . 2 {⟨𝑎, 𝑧⟩ ∣ 𝜓} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
4341, 42eqtr3i 2748 1 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1596  ∃wex 1817  Ⅎwnf 1821   ∈ wcel 2103  {cab 2710  Vcvv 3304  ⟨cop 4291  {copab 4820   × cxp 5216  ◡ccnv 5217  ‘cfv 6001  {coprab 6766  1st c1st 7283  2nd c2nd 7284 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-iota 5964  df-fun 6003  df-fv 6009  df-oprab 6769  df-1st 7285  df-2nd 7286 This theorem is referenced by: (None)
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