Theorem brcupg 5815

Description: Binary relationship form of the cup function. (Contributed by SF, 11-Feb-2015.)

Assertion

Ref	Expression
brcupg	⊢ ((A ∈ V ∧ B ∈ W) → (⟨A, B⟩ Cup C ↔ C = (A ∪ B)))

Proof of Theorem brcupg

Step	Hyp	Ref	Expression
1		fncup 5814	. . 3 ⊢ Cup Fn V
2		opexg 4588	. . 3 ⊢ ((A ∈ V ∧ B ∈ W) → ⟨A, B⟩ ∈ V)
3		fnbrfvb 5359	. . 3 ⊢ (( Cup Fn V ∧ ⟨A, B⟩ ∈ V) → (( Cup ‘⟨A, B⟩) = C ↔ ⟨A, B⟩ Cup C))
4	1, 2, 3	sylancr 644	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → (( Cup ‘⟨A, B⟩) = C ↔ ⟨A, B⟩ Cup C))
5		cupvalg 5813	. . . 4 ⊢ ((A ∈ V ∧ B ∈ W) → (A Cup B) = (A ∪ B))
6	5	eqeq1d 2361	. . 3 ⊢ ((A ∈ V ∧ B ∈ W) → ((A Cup B) = C ↔ (A ∪ B) = C))
7		df-ov 5527	. . . 4 ⊢ (A Cup B) = ( Cup ‘⟨A, B⟩)
8	7	eqeq1i 2360	. . 3 ⊢ ((A Cup B) = C ↔ ( Cup ‘⟨A, B⟩) = C)
9		eqcom 2355	. . 3 ⊢ ((A ∪ B) = C ↔ C = (A ∪ B))
10	6, 8, 9	3bitr3g 278	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → (( Cup ‘⟨A, B⟩) = C ↔ C = (A ∪ B)))
11	4, 10	bitr3d 246	1 ⊢ ((A ∈ V ∧ B ∈ W) → (⟨A, B⟩ Cup C ↔ C = (A ∪ B)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∪ cun 3208  ⟨cop 4562   class class class wbr 4640   Fn wfn 4777   ‘cfv 4782  (class class class)co 5526   Cup ccup 5742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-fo 4794  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-cup 5743

This theorem is referenced by:  brcup  5816