Theorem oprabval2 3051

Description: The value of an operation abstraction. Special case.

Hypotheses

Ref	Expression
oprabval2.1	⊢ S ∈ V
oprabval2.2	⊢ (x = A → R = G)
oprabval2.3	⊢ (y = B → G = S)
oprabval2.4	⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ C ∧ y ∈ D) ∧ z = R)}

Assertion

Ref	Expression
oprabval2	⊢ ((A ∈ C ∧ B ∈ D) → (AFB) = S)

Distinct variable group(s):   x,y,z,A   x,B,y,z   x,C,y,z   x,D,y,z   x,G   z,R   x,S,y,z

Proof of Theorem oprabval2

Step	Hyp	Ref	Expression
1		oprabval2.1	. 2 ⊢ S ∈ V
2		oprabval2.2	. . 3 ⊢ (x = A → R = G)
3		oprabval2.3	. . 3 ⊢ (y = B → G = S)
4		oprabval2.4	. . 3 ⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ C ∧ y ∈ D) ∧ z = R)}
5	2, 3, 4	oprabval2g 3050	. 2 ⊢ ((A ∈ C ∧ B ∈ D ∧ S ∈ V) → (AFB) = S)
6	1, 5	mp3an3 641	1 ⊢ ((A ∈ C ∧ B ∈ D) → (AFB) = S)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  (class class class)co 3001  {copab2 3002

This theorem is referenced by:  1st2val 3097  oav 3119  omv 3120  oev 3122  genpv 3896  subval 4134  divval 4217  seqval 4665  expvalt 4677  ruclem15 4899  hvsubvalt 4997  shsumvalt 5279  sshjvalt 5321  sshjval3t 5327  hosmvalt 5487  hodmvalt 5488

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004

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