Theorem bj-opabco 37927

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 5668	. 2 ⊢ ({⟨𝑦, 𝑧⟩ ∣ 𝜓} ∘ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)}
2		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑥𝑎
3		nfopab1 5179	. . . . . 6 ⊢ Ⅎ𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
4		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑥𝑐
5	2, 3, 4	nfbr 5156	. . . . 5 ⊢ Ⅎ𝑥 𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐
6		nfv 1947	. . . . 5 ⊢ Ⅎ𝑥 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏
7	5, 6	nfan 1932	. . . 4 ⊢ Ⅎ𝑥(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
8	7	nfex 2356	. . 3 ⊢ Ⅎ𝑥∃𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
9		nfv 1947	. . . . 5 ⊢ Ⅎ𝑧 𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐
10		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑧𝑐
11		nfopab2 5180	. . . . . 6 ⊢ Ⅎ𝑧{⟨𝑦, 𝑧⟩ ∣ 𝜓}
12		nfcv 2924	. . . . . 6 ⊢ Ⅎ𝑧𝑏
13	10, 11, 12	nfbr 5156	. . . . 5 ⊢ Ⅎ𝑧 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏
14	9, 13	nfan 1932	. . . 4 ⊢ Ⅎ𝑧(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
15	14	nfex 2356	. . 3 ⊢ Ⅎ𝑧∃𝑐(𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ∧ 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏)
16		nfv 1947	. . 3 ⊢ Ⅎ𝑎∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)
17		nfv 1947	. . 3 ⊢ Ⅎ𝑏∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)
18		nfv 1947	. . . 4 ⊢ Ⅎ𝑦(𝑎 = 𝑥 ∧ 𝑏 = 𝑧)
19		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑦𝑎
20		nfopab2 5180	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
21		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑦𝑐
22	19, 20, 21	nfbr 5156	. . . . . 6 ⊢ Ⅎ𝑦 𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐
23		nfopab1 5179	. . . . . . 7 ⊢ Ⅎ𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}
24		nfcv 2924	. . . . . . 7 ⊢ Ⅎ𝑦𝑏
25	21, 23, 24	nfbr 5156	. . . . . 6 ⊢ Ⅎ𝑦 𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏
26	22, 25	nfan 1932	. . . . 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 779	. . . . . . 7 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → 𝑎 = 𝑥)
29		simpr 490	. . . . . . 7 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → 𝑐 = 𝑦)
30	28, 29	breq12d 5120	. . . . . 6 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → (𝑎{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑐 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦))
31		simplr 781	. . . . . . 7 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → 𝑏 = 𝑧)
32	29, 31	breq12d 5120	. . . . . 6 ⊢ (((𝑎 = 𝑥 ∧ 𝑏 = 𝑧) ∧ 𝑐 = 𝑦) → (𝑐{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑏 ↔ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧))
33	30, 32	anbi12d 644	. . . . 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 418	. . . 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 2370	. . 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 5181	. 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 37926	. . . . 5 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ 𝜑)
38		bj-opelopabid 37926	. . . . 5 ⊢ (𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧 ↔ 𝜓)
39	37, 38	anbi12i 640	. . . 4 ⊢ ((𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧) ↔ (𝜑 ∧ 𝜓))
40	39	exbii 1881	. . 3 ⊢ (∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧) ↔ ∃𝑦(𝜑 ∧ 𝜓))
41	40	opabbii 5176	. 2 ⊢ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ∧ 𝑦{⟨𝑦, 𝑧⟩ ∣ 𝜓}𝑧)} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝜑 ∧ 𝜓)}
42	1, 36, 41	3eqtri 2789	1 ⊢ ({⟨𝑦, 𝑧⟩ ∣ 𝜓} ∘ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝜑 ∧ 𝜓)}

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812  Ⅎwnf 1816   class class class wbr 5107  {copab 5171   ∘ ccom 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-co 5668

This theorem is used by:  bj-imdirco  37929