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Theorem bj-opabco 34598
Description: Composition of ordered-pair class abstractions. (Contributed by BJ, 22-May-2024.)
Assertion
Ref Expression
bj-opabco ({⟨𝑦, 𝑧⟩ ∣ 𝜓} ∘ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝜑𝜓)}
Distinct variable groups:   𝑥,𝑦,𝑧   𝜓,𝑥   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦,𝑧)

Proof of Theorem bj-opabco
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-co 5532 . 2 ({⟨𝑦, 𝑧⟩ ∣ 𝜓} ∘ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)}
2 nfcv 2958 . . . . . 6 𝑥𝑎
3 nfopab1 5102 . . . . . 6 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
4 nfcv 2958 . . . . . 6 𝑥𝑐
52, 3, 4nfbr 5080 . . . . 5 𝑥 𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐
6 nfv 1915 . . . . 5 𝑥 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏
75, 6nfan 1900 . . . 4 𝑥(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
87nfex 2335 . . 3 𝑥𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
9 nfv 1915 . . . . 5 𝑧 𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐
10 nfcv 2958 . . . . . 6 𝑧𝑐
11 nfopab2 5103 . . . . . 6 𝑧{⟨𝑦, 𝑧⟩ ∣ 𝜓}
12 nfcv 2958 . . . . . 6 𝑧𝑏
1310, 11, 12nfbr 5080 . . . . 5 𝑧 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏
149, 13nfan 1900 . . . 4 𝑧(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
1514nfex 2335 . . 3 𝑧𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
16 nfv 1915 . . 3 𝑎𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)
17 nfv 1915 . . 3 𝑏𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)
18 nfv 1915 . . . 4 𝑦(𝑎 = 𝑥𝑏 = 𝑧)
19 nfcv 2958 . . . . . . 7 𝑦𝑎
20 nfopab2 5103 . . . . . . 7 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
21 nfcv 2958 . . . . . . 7 𝑦𝑐
2219, 20, 21nfbr 5080 . . . . . 6 𝑦 𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐
23 nfopab1 5102 . . . . . . 7 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}
24 nfcv 2958 . . . . . . 7 𝑦𝑏
2521, 23, 24nfbr 5080 . . . . . 6 𝑦 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏
2622, 25nfan 1900 . . . . 5 𝑦(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
2726a1i 11 . . . 4 ((𝑎 = 𝑥𝑏 = 𝑧) → Ⅎ𝑦(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏))
28 simpll 766 . . . . . . 7 (((𝑎 = 𝑥𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → 𝑎 = 𝑥)
29 simpr 488 . . . . . . 7 (((𝑎 = 𝑥𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → 𝑐 = 𝑦)
3028, 29breq12d 5046 . . . . . 6 (((𝑎 = 𝑥𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → (𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦))
31 simplr 768 . . . . . . 7 (((𝑎 = 𝑥𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → 𝑏 = 𝑧)
3229, 31breq12d 5046 . . . . . 6 (((𝑎 = 𝑥𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → (𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧))
3330, 32anbi12d 633 . . . . 5 (((𝑎 = 𝑥𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → ((𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏) ↔ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)))
3433ex 416 . . . 4 ((𝑎 = 𝑥𝑏 = 𝑧) → (𝑐 = 𝑦 → ((𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏) ↔ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧))))
3518, 27, 34cbvexdw 2351 . . 3 ((𝑎 = 𝑥𝑏 = 𝑧) → (∃𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏) ↔ ∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)))
368, 15, 16, 17, 35cbvopab 5104 . 2 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)}
37 bj-opelopabid 34597 . . . . 5 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝜑)
38 bj-opelopabid 34597 . . . . 5 (𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧𝜓)
3937, 38anbi12i 629 . . . 4 ((𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧) ↔ (𝜑𝜓))
4039exbii 1849 . . 3 (∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧) ↔ ∃𝑦(𝜑𝜓))
4140opabbii 5100 . 2 {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝜑𝜓)}
421, 36, 413eqtri 2828 1 ({⟨𝑦, 𝑧⟩ ∣ 𝜓} ∘ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝜑𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wex 1781  wnf 1785   class class class wbr 5033  {copab 5095  ccom 5527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-co 5532
This theorem is referenced by:  bj-imdirco  34600
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