Theorem opelco3 36202

Description: Alternate way of saying that an ordered pair is in a composition. (Contributed by Scott Fenton, 6-May-2018.)

Assertion

Ref	Expression
opelco3	⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))

Proof of Theorem opelco3

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 5114	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷))
2		relco 6113	. . . 4 ⊢ Rel (𝐶 ∘ 𝐷)
3	2	brrelex12i 5719	. . 3 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		snprc 4688	. . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
5		noel 4299	. . . . . . 7 ⊢ ¬ 𝐵 ∈ ∅
6		imaeq2 6061	. . . . . . . . . 10 ⊢ ({𝐴} = ∅ → (𝐷 “ {𝐴}) = (𝐷 “ ∅))
7	6	imaeq2d 6065	. . . . . . . . 9 ⊢ ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = (𝐶 “ (𝐷 “ ∅)))
8		ima0 6082	. . . . . . . . . . 11 ⊢ (𝐷 “ ∅) = ∅
9	8	imaeq2i 6063	. . . . . . . . . 10 ⊢ (𝐶 “ (𝐷 “ ∅)) = (𝐶 “ ∅)
10		ima0 6082	. . . . . . . . . 10 ⊢ (𝐶 “ ∅) = ∅
11	9, 10	eqtri 2792	. . . . . . . . 9 ⊢ (𝐶 “ (𝐷 “ ∅)) = ∅
12	7, 11	eqtrdi 2820	. . . . . . . 8 ⊢ ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = ∅)
13	12	eleq2d 2855	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ 𝐵 ∈ ∅))
14	5, 13	mtbiri 330	. . . . . 6 ⊢ ({𝐴} = ∅ → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
15	4, 14	sylbi 220	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
16	15	con4i 115	. . . 4 ⊢ (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐴 ∈ V)
17		elex 3484	. . . 4 ⊢ (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐵 ∈ V)
18	16, 17	jca 520	. . 3 ⊢ (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
19		df-rex 3096	. . . . 5 ⊢ (∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵 ↔ ∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵))
20		elimasng 6094	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
21	20	elvd 3469	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
22		df-br 5114	. . . . . . . . 9 ⊢ (𝐴𝐷𝑧 ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷)
23	21, 22	bitr4di 292	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
24	23	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
25	24	anbi1d 642	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ (𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
26	25	exbidv 1948	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ ∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
27	19, 26	bitr2id 287	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
28		brcog 5855	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
29		elimag 6069	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
30	29	adantl 486	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
31	27, 28, 30	3bitr4d 314	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴}))))
32	3, 18, 31	pm5.21nii 381	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
33	1, 32	bitr3i 280	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃wrex 3095  Vcvv 3463  ∅c0 4294  {csn 4594  ⟨cop 4600   class class class wbr 5113   “ cima 5667   ∘ ccom 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5407

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677

This theorem is referenced by: (None)