Theorem bj-opabco 34868

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(𝜑 ∧ 𝜓)}

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.

Step	Hyp	Ref	Expression
1		df-co 5526	. 2 ⊢ ({⟨𝑦, 𝑧⟩ ∣ 𝜓} ∘ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)}
2		nfcv 2917	. . . . . 6 ⊢ Ⅎ𝑥𝑎
3		nfopab1 5094	. . . . . 6 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
4		nfcv 2917	. . . . . 6 ⊢ Ⅎ𝑥𝑐
5	2, 3, 4	nfbr 5072	. . . . 5 ⊢ Ⅎ𝑥 𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐
6		nfv 1916	. . . . 5 ⊢ Ⅎ𝑥 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏
7	5, 6	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
8	7	nfex 2333	. . 3 ⊢ Ⅎ𝑥∃𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
9		nfv 1916	. . . . 5 ⊢ Ⅎ𝑧 𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐
10		nfcv 2917	. . . . . 6 ⊢ Ⅎ𝑧𝑐
11		nfopab2 5095	. . . . . 6 ⊢ Ⅎ𝑧{⟨𝑦, 𝑧⟩ ∣ 𝜓}
12		nfcv 2917	. . . . . 6 ⊢ Ⅎ𝑧𝑏
13	10, 11, 12	nfbr 5072	. . . . 5 ⊢ Ⅎ𝑧 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏
14	9, 13	nfan 1901	. . . 4 ⊢ Ⅎ𝑧(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
15	14	nfex 2333	. . 3 ⊢ Ⅎ𝑧∃𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
16		nfv 1916	. . 3 ⊢ Ⅎ𝑎∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)
17		nfv 1916	. . 3 ⊢ Ⅎ𝑏∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)
18		nfv 1916	. . . 4 ⊢ Ⅎ𝑦(𝑎 = 𝑥 ∧ 𝑏 = 𝑧)
19		nfcv 2917	. . . . . . 7 ⊢ Ⅎ𝑦𝑎
20		nfopab2 5095	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
21		nfcv 2917	. . . . . . 7 ⊢ Ⅎ𝑦𝑐
22	19, 20, 21	nfbr 5072	. . . . . 6 ⊢ Ⅎ𝑦 𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐
23		nfopab1 5094	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}
24		nfcv 2917	. . . . . . 7 ⊢ Ⅎ𝑦𝑏
25	21, 23, 24	nfbr 5072	. . . . . 6 ⊢ Ⅎ𝑦 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏
26	22, 25	nfan 1901	. . . . 5 ⊢ Ⅎ𝑦(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
27	26	a1i 11	. . . 4 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) → Ⅎ𝑦(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏))
28		simpll 767	. . . . . . 7 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → 𝑎 = 𝑥)
29		simpr 489	. . . . . . 7 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → 𝑐 = 𝑦)
30	28, 29	breq12d 5038	. . . . . 6 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → (𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦))
31		simplr 769	. . . . . . 7 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → 𝑏 = 𝑧)
32	29, 31	breq12d 5038	. . . . . 6 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → (𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏 ↔ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧))
33	30, 32	anbi12d 634	. . . . 5 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → ((𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏) ↔ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)))
34	33	ex 417	. . . 4 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) → (𝑐 = 𝑦 → ((𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏) ↔ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧))))
35	18, 27, 34	cbvexdw 2349	. . 3 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) → (∃𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏) ↔ ∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)))
36	8, 15, 16, 17, 35	cbvopab 5096	. 2 ⊢ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)}
37		bj-opelopabid 34867	. . . . 5 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑)
38		bj-opelopabid 34867	. . . . 5 ⊢ (𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧 ↔ 𝜓)
39	37, 38	anbi12i 630	. . . 4 ⊢ ((𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧) ↔ (𝜑 ∧ 𝜓))
40	39	exbii 1850	. . 3 ⊢ (∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧) ↔ ∃𝑦(𝜑 ∧ 𝜓))
41	40	opabbii 5092	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝜑 ∧ 𝜓)}
42	1, 36, 41	3eqtri 2786	1 ⊢ ({⟨𝑦, 𝑧⟩ ∣ 𝜓} ∘ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝜑 ∧ 𝜓)}

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1539  ∃wex 1782  Ⅎwnf 1786   class class class wbr 5025  {copab 5087   ∘ ccom 5521

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-v 3409  df-dif 3857  df-un 3859  df-nul 4222  df-if 4414  df-sn 4516  df-pr 4518  df-op 4522  df-br 5026  df-opab 5088  df-co 5526

This theorem is referenced by:  bj-imdirco  34870