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Theorem opelco3 35756
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 6129 . . . 4 Rel (𝐶𝐷)
32brrelex12i 5744 . . 3 (𝐴(𝐶𝐷)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 snprc 4722 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
5 noel 4344 . . . . . . 7 ¬ 𝐵 ∈ ∅
6 imaeq2 6076 . . . . . . . . . 10 ({𝐴} = ∅ → (𝐷 “ {𝐴}) = (𝐷 “ ∅))
76imaeq2d 6080 . . . . . . . . 9 ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = (𝐶 “ (𝐷 “ ∅)))
8 ima0 6097 . . . . . . . . . . 11 (𝐷 “ ∅) = ∅
98imaeq2i 6078 . . . . . . . . . 10 (𝐶 “ (𝐷 “ ∅)) = (𝐶 “ ∅)
10 ima0 6097 . . . . . . . . . 10 (𝐶 “ ∅) = ∅
119, 10eqtri 2763 . . . . . . . . 9 (𝐶 “ (𝐷 “ ∅)) = ∅
127, 11eqtrdi 2791 . . . . . . . 8 ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = ∅)
1312eleq2d 2825 . . . . . . 7 ({𝐴} = ∅ → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ 𝐵 ∈ ∅))
145, 13mtbiri 327 . . . . . 6 ({𝐴} = ∅ → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
154, 14sylbi 217 . . . . 5 𝐴 ∈ V → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
1615con4i 114 . . . 4 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐴 ∈ V)
17 elex 3499 . . . 4 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐵 ∈ V)
1816, 17jca 511 . . 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 6109 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝑧 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
2120elvd 3484 . . . . . . . . 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 631 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ (𝐴𝐷𝑧𝑧𝐶𝐵)))
2625exbidv 1919 . . . . 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 5880 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵)))
29 elimag 6084 . . . . 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 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wrex 3068  Vcvv 3478  c0 4339  {csn 4631  cop 4637   class class class wbr 5148  cima 5692  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by: (None)
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