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Theorem opelco3 32915
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 5058 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷))
2 relco 6090 . . . 4 Rel (𝐶𝐷)
32brrelex12i 5600 . . 3 (𝐴(𝐶𝐷)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
4 snprc 4645 . . . . . 6 𝐴 ∈ V ↔ {𝐴} = ∅)
5 noel 4293 . . . . . . 7 ¬ 𝐵 ∈ ∅
6 imaeq2 5918 . . . . . . . . . 10 ({𝐴} = ∅ → (𝐷 “ {𝐴}) = (𝐷 “ ∅))
76imaeq2d 5922 . . . . . . . . 9 ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = (𝐶 “ (𝐷 “ ∅)))
8 ima0 5938 . . . . . . . . . . 11 (𝐷 “ ∅) = ∅
98imaeq2i 5920 . . . . . . . . . 10 (𝐶 “ (𝐷 “ ∅)) = (𝐶 “ ∅)
10 ima0 5938 . . . . . . . . . 10 (𝐶 “ ∅) = ∅
119, 10eqtri 2841 . . . . . . . . 9 (𝐶 “ (𝐷 “ ∅)) = ∅
127, 11syl6eq 2869 . . . . . . . 8 ({𝐴} = ∅ → (𝐶 “ (𝐷 “ {𝐴})) = ∅)
1312eleq2d 2895 . . . . . . 7 ({𝐴} = ∅ → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ 𝐵 ∈ ∅))
145, 13mtbiri 328 . . . . . 6 ({𝐴} = ∅ → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
154, 14sylbi 218 . . . . 5 𝐴 ∈ V → ¬ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
1615con4i 114 . . . 4 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐴 ∈ V)
17 elex 3510 . . . 4 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → 𝐵 ∈ V)
1816, 17jca 512 . . 3 (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
19 df-rex 3141 . . . . 5 (∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵 ↔ ∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵))
20 elimasng 5948 . . . . . . . . . 10 ((𝐴 ∈ V ∧ 𝑧 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
2120elvd 3498 . . . . . . . . 9 (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷))
22 df-br 5058 . . . . . . . . 9 (𝐴𝐷𝑧 ↔ ⟨𝐴, 𝑧⟩ ∈ 𝐷)
2321, 22syl6bbr 290 . . . . . . . 8 (𝐴 ∈ V → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
2423adantr 481 . . . . . . 7 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑧 ∈ (𝐷 “ {𝐴}) ↔ 𝐴𝐷𝑧))
2524anbi1d 629 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ (𝐴𝐷𝑧𝑧𝐶𝐵)))
2625exbidv 1913 . . . . 5 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝑧 ∈ (𝐷 “ {𝐴}) ∧ 𝑧𝐶𝐵) ↔ ∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵)))
2719, 26syl5rbb 285 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
28 brcog 5730 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑧(𝐴𝐷𝑧𝑧𝐶𝐵)))
29 elimag 5926 . . . . 5 (𝐵 ∈ V → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
3029adantl 482 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})) ↔ ∃𝑧 ∈ (𝐷 “ {𝐴})𝑧𝐶𝐵))
3127, 28, 303bitr4d 312 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴}))))
323, 18, 31pm5.21nii 380 . 2 (𝐴(𝐶𝐷)𝐵𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
331, 32bitr3i 278 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ 𝐵 ∈ (𝐶 “ (𝐷 “ {𝐴})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wrex 3136  Vcvv 3492  c0 4288  {csn 4557  cop 4563   class class class wbr 5057  cima 5551  ccom 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561
This theorem is referenced by: (None)
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