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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⟩ ∈ (𝐶𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))

Proof of Theorem opelco3
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-br 4584 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷))
2 relco 5550 . . . 4 Rel (𝐶𝐷)
3 brrelex12 5079 . . . 4 ((Rel (𝐶𝐷) ∧ 𝐴(𝐶𝐷)𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
42, 3mpan 702 . . 3 (𝐴(𝐶𝐷)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 “ ∅))
87imaeq2d 5385 . . . . . . . . 9 ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = (𝐶 “ (𝐷 “ ∅)))
9 ima0 5400 . . . . . . . . . . 11 (𝐷 “ ∅) = ∅
109imaeq2i 5383 . . . . . . . . . 10 (𝐶 “ (𝐷 “ ∅)) = (𝐶 “ ∅)
11 ima0 5400 . . . . . . . . . 10 (𝐶 “ ∅) = ∅
1210, 11eqtri 2632 . . . . . . . . 9 (𝐶 “ (𝐷 “ ∅)) = ∅
138, 12syl6eq 2660 . . . . . . . 8 ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = ∅)
1413eleq2d 2673 . . . . . . 7 ({𝐴} = ∅ → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ 𝐵 ∈ ∅))
156, 14mtbiri 316 . . . . . 6 ({𝐴} = ∅ → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
165, 15sylbi 206 . . . . 5 𝐴 ∈ V → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
1716con4i 112 . . . 4 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐴 ∈ V)
18 elex 3185 . . . 4 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐵 ∈ V)
1917, 18jca 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))
2321, 22mpan2 703 . . . . . . . . 9 (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
24 df-br 4584 . . . . . . . . 9 (𝐴𝐷𝑧 ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷)
2523, 24syl6bbr 277 . . . . . . . 8 (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
2625adantr 480 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
2726anbi1d 737 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ (𝐴𝐷𝑧𝑧𝐶𝐵)))
2827exbidv 1837 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ ∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵)))
2920, 28syl5rbb 272 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
30 brcog 5210 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵)))
31 elimag 5389 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
3231adantl 481 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
3329, 30, 323bitr4d 299 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴}))))
344, 19, 33pm5.21nii 367 . 2 (𝐴(𝐶𝐷)𝐵𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
351, 34bitr3i 265 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
 Colors of variables: wff setvar class 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)
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