Theorem opelco3 30923

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 4584	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷))
2		relco 5550	. . . 4 ⊢ Rel (𝐶 ∘ 𝐷)
3		brrelex12 5079	. . . 4 ⊢ ((Rel (𝐶 ∘ 𝐷) ∧ 𝐴(𝐶 ∘ 𝐷)𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4	2, 3	mpan 702	. . 3 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5		snprc 4197	. . . . . 6 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
6		noel 3878	. . . . . . 7 ⊢ ¬ 𝐵 ∈ ∅
7		imaeq2 5381	. . . . . . . . . 10 ⊢ ({𝐴} = ∅ → (𝐷 “ {𝐴}) = (𝐷 “ ∅))
8	7	imaeq2d 5385	. . . . . . . . 9 ⊢ ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = (𝐶 “ (𝐷 “ ∅)))
9		ima0 5400	. . . . . . . . . . 11 ⊢ (𝐷 “ ∅) = ∅
10	9	imaeq2i 5383	. . . . . . . . . 10 ⊢ (𝐶 “ (𝐷 “ ∅)) = (𝐶 “ ∅)
11		ima0 5400	. . . . . . . . . 10 ⊢ (𝐶 “ ∅) = ∅
12	10, 11	eqtri 2632	. . . . . . . . 9 ⊢ (𝐶 “ (𝐷 “ ∅)) = ∅
13	8, 12	syl6eq 2660	. . . . . . . 8 ⊢ ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = ∅)
14	13	eleq2d 2673	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ 𝐵 ∈ ∅))
15	6, 14	mtbiri 316	. . . . . 6 ⊢ ({𝐴} = ∅ → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
16	5, 15	sylbi 206	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
17	16	con4i 112	. . . 4 ⊢ (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐴 ∈ V)
18		elex 3185	. . . 4 ⊢ (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐵 ∈ V)
19	17, 18	jca 553	. . 3 ⊢ (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
20		df-rex 2902	. . . . 5 ⊢ (∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵 ↔ ∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵))
21		vex 3176	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
22		elimasng 5410	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
23	21, 22	mpan2 703	. . . . . . . . 9 ⊢ (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
24		df-br 4584	. . . . . . . . 9 ⊢ (𝐴𝐷𝑧 ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷)
25	23, 24	syl6bbr 277	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
26	25	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
27	26	anbi1d 737	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ (𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
28	27	exbidv 1837	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ ∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
29	20, 28	syl5rbb 272	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
30		brcog 5210	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑧(𝐴𝐷𝑧 ∧ 𝑧𝐶𝐵)))
31		elimag 5389	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
32	31	adantl 481	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
33	29, 30, 32	3bitr4d 299	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴}))))
34	4, 19, 33	pm5.21nii 367	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
35	1, 34	bitr3i 265	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))

Syntax hints:  ¬ wn 3   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583   “ cima 5041   ∘ ccom 5042  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051

This theorem is referenced by: (None)