Theorem opelco3 35050

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 5148	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷))
2		relco 6106	. . . 4 ⊢ Rel (𝐶 ∘ 𝐷)
3	2	brrelex12i 5730	. . 3 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		snprc 4720	. . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
5		noel 4329	. . . . . . 7 ⊢ ¬ 𝐵 ∈ ∅
6		imaeq2 6054	. . . . . . . . . 10 ⊢ ({𝐴} = ∅ → (𝐷 “ {𝐴}) = (𝐷 “ ∅))
7	6	imaeq2d 6058	. . . . . . . . 9 ⊢ ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = (𝐶 “ (𝐷 “ ∅)))
8		ima0 6075	. . . . . . . . . . 11 ⊢ (𝐷 “ ∅) = ∅
9	8	imaeq2i 6056	. . . . . . . . . 10 ⊢ (𝐶 “ (𝐷 “ ∅)) = (𝐶 “ ∅)
10		ima0 6075	. . . . . . . . . 10 ⊢ (𝐶 “ ∅) = ∅
11	9, 10	eqtri 2758	. . . . . . . . 9 ⊢ (𝐶 “ (𝐷 “ ∅)) = ∅
12	7, 11	eqtrdi 2786	. . . . . . . 8 ⊢ ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = ∅)
13	12	eleq2d 2817	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ 𝐵 ∈ ∅))
14	5, 13	mtbiri 326	. . . . . 6 ⊢ ({𝐴} = ∅ → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
15	4, 14	sylbi 216	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
16	15	con4i 114	. . . 4 ⊢ (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐴 ∈ V)
17		elex 3491	. . . 4 ⊢ (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐵 ∈ V)
18	16, 17	jca 510	. . 3 ⊢ (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
19		df-rex 3069	. . . . 5 ⊢ (∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵 ↔ ∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵))
20		elimasng 6086	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
21	20	elvd 3479	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
22		df-br 5148	. . . . . . . . 9 ⊢ (𝐴𝐷𝑧 ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷)
23	21, 22	bitr4di 288	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
24	23	adantr 479	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
25	24	anbi1d 628	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ (𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
26	25	exbidv 1922	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ ∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
27	19, 26	bitr2id 283	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
28		brcog 5865	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
29		elimag 6062	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
30	29	adantl 480	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
31	27, 28, 30	3bitr4d 310	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴}))))
32	3, 18, 31	pm5.21nii 377	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
33	1, 32	bitr3i 276	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 394   = wceq 1539  ∃wex 1779   ∈ wcel 2104  ∃wrex 3068  Vcvv 3472  ∅c0 4321  {csn 4627  ⟨cop 4633   class class class wbr 5147   “ cima 5678   ∘ ccom 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688

This theorem is referenced by: (None)