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Theorem opelco3 31433
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 4624 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷))
2 relco 5602 . . . 4 Rel (𝐶𝐷)
3 brrelex12 5125 . . . 4 ((Rel (𝐶𝐷) ∧ 𝐴(𝐶𝐷)𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
42, 3mpan 705 . . 3 (𝐴(𝐶𝐷)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
5 snprc 4230 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
6 noel 3901 . . . . . . 7 ¬ 𝐵 ∈ ∅
7 imaeq2 5431 . . . . . . . . . 10 ({𝐴} = ∅ → (𝐷 “ {𝐴}) = (𝐷 “ ∅))
87imaeq2d 5435 . . . . . . . . 9 ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = (𝐶 “ (𝐷 “ ∅)))
9 ima0 5450 . . . . . . . . . . 11 (𝐷 “ ∅) = ∅
109imaeq2i 5433 . . . . . . . . . 10 (𝐶 “ (𝐷 “ ∅)) = (𝐶 “ ∅)
11 ima0 5450 . . . . . . . . . 10 (𝐶 “ ∅) = ∅
1210, 11eqtri 2643 . . . . . . . . 9 (𝐶 “ (𝐷 “ ∅)) = ∅
138, 12syl6eq 2671 . . . . . . . 8 ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = ∅)
1413eleq2d 2684 . . . . . . 7 ({𝐴} = ∅ → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ 𝐵 ∈ ∅))
156, 14mtbiri 317 . . . . . 6 ({𝐴} = ∅ → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
165, 15sylbi 207 . . . . 5 𝐴 ∈ V → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
1716con4i 113 . . . 4 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐴 ∈ V)
18 elex 3202 . . . 4 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐵 ∈ V)
1917, 18jca 554 . . 3 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
20 df-rex 2914 . . . . 5 (∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵 ↔ ∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵))
21 vex 3193 . . . . . . . . . 10 𝑧 ∈ V
22 elimasng 5460 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝑧 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
2321, 22mpan2 706 . . . . . . . . 9 (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
24 df-br 4624 . . . . . . . . 9 (𝐴𝐷𝑧 ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷)
2523, 24syl6bbr 278 . . . . . . . 8 (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
2625adantr 481 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
2726anbi1d 740 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ (𝐴𝐷𝑧𝑧𝐶𝐵)))
2827exbidv 1847 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ ∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵)))
2920, 28syl5rbb 273 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
30 brcog 5258 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵)))
31 elimag 5439 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
3231adantl 482 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
3329, 30, 323bitr4d 300 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴}))))
344, 19, 33pm5.21nii 368 . 2 (𝐴(𝐶𝐷)𝐵𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
351, 34bitr3i 266 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2909  Vcvv 3190  c0 3897  {csn 4155  cop 4161   class class class wbr 4623  cima 5087  ccom 5088  Rel wrel 5089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097
This theorem is referenced by: (None)
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