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Theorem brcup 33404
 Description: Binary relation form of the Cup function. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brcup.1 𝐴 ∈ V
brcup.2 𝐵 ∈ V
brcup.3 𝐶 ∈ V
Assertion
Ref Expression
brcup (⟨𝐴, 𝐵⟩Cup𝐶𝐶 = (𝐴𝐵))

Proof of Theorem brcup
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opex 5328 . 2 𝐴, 𝐵⟩ ∈ V
2 brcup.3 . 2 𝐶 ∈ V
3 df-cup 33334 . 2 Cup = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (((1st ∘ E ) ∪ (2nd ∘ E )) ⊗ V)))
4 brcup.1 . . . 4 𝐴 ∈ V
5 brcup.2 . . . 4 𝐵 ∈ V
64, 5opelvv 5566 . . 3 𝐴, 𝐵⟩ ∈ (V × V)
7 brxp 5573 . . 3 (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
86, 2, 7mpbir2an 709 . 2 𝐴, 𝐵⟩((V × V) × V)𝐶
9 epel 5441 . . . . . . 7 (𝑥 E 𝑦𝑥𝑦)
10 vex 3473 . . . . . . . . 9 𝑦 ∈ V
1110, 1brcnv 5725 . . . . . . . 8 (𝑦1st𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩1st 𝑦)
124, 5br1steq 33018 . . . . . . . 8 (⟨𝐴, 𝐵⟩1st 𝑦𝑦 = 𝐴)
1311, 12bitri 277 . . . . . . 7 (𝑦1st𝐴, 𝐵⟩ ↔ 𝑦 = 𝐴)
149, 13anbi12ci 629 . . . . . 6 ((𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩) ↔ (𝑦 = 𝐴𝑥𝑦))
1514exbii 1848 . . . . 5 (∃𝑦(𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
16 vex 3473 . . . . . 6 𝑥 ∈ V
1716, 1brco 5713 . . . . 5 (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦𝑦1st𝐴, 𝐵⟩))
184clel3 3631 . . . . 5 (𝑥𝐴 ↔ ∃𝑦(𝑦 = 𝐴𝑥𝑦))
1915, 17, 183bitr4i 305 . . . 4 (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥𝐴)
2010, 1brcnv 5725 . . . . . . . 8 (𝑦2nd𝐴, 𝐵⟩ ↔ ⟨𝐴, 𝐵⟩2nd 𝑦)
214, 5br2ndeq 33019 . . . . . . . 8 (⟨𝐴, 𝐵⟩2nd 𝑦𝑦 = 𝐵)
2220, 21bitri 277 . . . . . . 7 (𝑦2nd𝐴, 𝐵⟩ ↔ 𝑦 = 𝐵)
239, 22anbi12ci 629 . . . . . 6 ((𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩) ↔ (𝑦 = 𝐵𝑥𝑦))
2423exbii 1848 . . . . 5 (∃𝑦(𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩) ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
2516, 1brco 5713 . . . . 5 (𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦(𝑥 E 𝑦𝑦2nd𝐴, 𝐵⟩))
265clel3 3631 . . . . 5 (𝑥𝐵 ↔ ∃𝑦(𝑦 = 𝐵𝑥𝑦))
2724, 25, 263bitr4i 305 . . . 4 (𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩ ↔ 𝑥𝐵)
2819, 27orbi12i 911 . . 3 ((𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ∨ 𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩) ↔ (𝑥𝐴𝑥𝐵))
29 brun 5089 . . 3 (𝑥((1st ∘ E ) ∪ (2nd ∘ E ))⟨𝐴, 𝐵⟩ ↔ (𝑥(1st ∘ E )⟨𝐴, 𝐵⟩ ∨ 𝑥(2nd ∘ E )⟨𝐴, 𝐵⟩))
30 elun 4100 . . 3 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
3128, 29, 303bitr4ri 306 . 2 (𝑥 ∈ (𝐴𝐵) ↔ 𝑥((1st ∘ E ) ∪ (2nd ∘ E ))⟨𝐴, 𝐵⟩)
321, 2, 3, 8, 31brtxpsd3 33361 1 (⟨𝐴, 𝐵⟩Cup𝐶𝐶 = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   ∧ wa 398   ∨ wo 843   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3470   ∪ cun 3907  ⟨cop 4545   class class class wbr 5038   E cep 5436   × cxp 5525  ◡ccnv 5526   ∘ ccom 5531  1st c1st 7661  2nd c2nd 7662  Cupccup 33311 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-symdif 4193  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-mpt 5119  df-id 5432  df-eprel 5437  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-fo 6333  df-fv 6335  df-1st 7663  df-2nd 7664  df-txp 33319  df-cup 33334 This theorem is referenced by:  brsuccf  33406
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