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Theorem opelco3 35051
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⟩ ∈ (𝐶𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))

Proof of Theorem opelco3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-br 5149 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷))
2 relco 6107 . . . 4 Rel (𝐶𝐷)
32brrelex12i 5731 . . 3 (𝐴(𝐶𝐷)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 snprc 4721 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
5 noel 4330 . . . . . . 7 ¬ 𝐵 ∈ ∅
6 imaeq2 6055 . . . . . . . . . 10 ({𝐴} = ∅ → (𝐷 “ {𝐴}) = (𝐷 “ ∅))
76imaeq2d 6059 . . . . . . . . 9 ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = (𝐶 “ (𝐷 “ ∅)))
8 ima0 6076 . . . . . . . . . . 11 (𝐷 “ ∅) = ∅
98imaeq2i 6057 . . . . . . . . . 10 (𝐶 “ (𝐷 “ ∅)) = (𝐶 “ ∅)
10 ima0 6076 . . . . . . . . . 10 (𝐶 “ ∅) = ∅
119, 10eqtri 2759 . . . . . . . . 9 (𝐶 “ (𝐷 “ ∅)) = ∅
127, 11eqtrdi 2787 . . . . . . . 8 ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = ∅)
1312eleq2d 2818 . . . . . . 7 ({𝐴} = ∅ → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ 𝐵 ∈ ∅))
145, 13mtbiri 327 . . . . . 6 ({𝐴} = ∅ → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
154, 14sylbi 216 . . . . 5 𝐴 ∈ V → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
1615con4i 114 . . . 4 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐴 ∈ V)
17 elex 3492 . . . 4 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐵 ∈ V)
1816, 17jca 511 . . 3 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
19 df-rex 3070 . . . . 5 (∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵 ↔ ∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵))
20 elimasng 6087 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝑧 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
2120elvd 3480 . . . . . . . . 9 (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
22 df-br 5149 . . . . . . . . 9 (𝐴𝐷𝑧 ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷)
2321, 22bitr4di 289 . . . . . . . 8 (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
2423adantr 480 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
2524anbi1d 629 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ (𝐴𝐷𝑧𝑧𝐶𝐵)))
2625exbidv 1923 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ ∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵)))
2719, 26bitr2id 284 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
28 brcog 5866 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵)))
29 elimag 6063 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
3029adantl 481 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
3127, 28, 303bitr4d 311 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴}))))
323, 18, 31pm5.21nii 378 . 2 (𝐴(𝐶𝐷)𝐵𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
331, 32bitr3i 277 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 395   = wceq 1540  wex 1780  wcel 2105  wrex 3069  Vcvv 3473  c0 4322  {csn 4628  cop 4634   class class class wbr 5148  cima 5679  ccom 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689
This theorem is referenced by: (None)
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