Theorem opelco3 35364

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 5143	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷))
2		relco 6106	. . . 4 ⊢ Rel (𝐶 ∘ 𝐷)
3	2	brrelex12i 5727	. . 3 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4		snprc 4717	. . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
5		noel 4326	. . . . . . 7 ⊢ ¬ 𝐵 ∈ ∅
6		imaeq2 6053	. . . . . . . . . 10 ⊢ ({𝐴} = ∅ → (𝐷 “ {𝐴}) = (𝐷 “ ∅))
7	6	imaeq2d 6057	. . . . . . . . 9 ⊢ ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = (𝐶 “ (𝐷 “ ∅)))
8		ima0 6074	. . . . . . . . . . 11 ⊢ (𝐷 “ ∅) = ∅
9	8	imaeq2i 6055	. . . . . . . . . 10 ⊢ (𝐶 “ (𝐷 “ ∅)) = (𝐶 “ ∅)
10		ima0 6074	. . . . . . . . . 10 ⊢ (𝐶 “ ∅) = ∅
11	9, 10	eqtri 2756	. . . . . . . . 9 ⊢ (𝐶 “ (𝐷 “ ∅)) = ∅
12	7, 11	eqtrdi 2784	. . . . . . . 8 ⊢ ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = ∅)
13	12	eleq2d 2815	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ 𝐵 ∈ ∅))
14	5, 13	mtbiri 327	. . . . . 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 3489	. . . 4 ⊢ (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐵 ∈ V)
18	16, 17	jca 511	. . 3 ⊢ (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
19		df-rex 3067	. . . . 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 3477	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
22		df-br 5143	. . . . . . . . 9 ⊢ (𝐴𝐷𝑧 ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷)
23	21, 22	bitr4di 289	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
24	23	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
25	24	anbi1d 630	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ (𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
26	25	exbidv 1917	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ ∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
27	19, 26	bitr2id 284	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
28		brcog 5863	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
29		elimag 6061	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
30	29	adantl 481	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
31	27, 28, 30	3bitr4d 311	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴}))))
32	3, 18, 31	pm5.21nii 378	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
33	1, 32	bitr3i 277	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∃wrex 3066  Vcvv 3470  ∅c0 4318  {csn 4624  ⟨cop 4630   class class class wbr 5142   “ cima 5675   ∘ ccom 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685

This theorem is referenced by: (None)