Theorem ofc2 7646

Description: Right operation by a constant. (Contributed by NM, 7-Oct-2014.)

Hypotheses

Ref	Expression
ofc2.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofc2.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)
ofc2.3	⊢ (𝜑 → 𝐹 Fn 𝐴)
ofc2.4	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)

Assertion

Ref	Expression
ofc2	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵))

Proof of Theorem ofc2

Step	Hyp	Ref	Expression
1		ofc2.3	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		ofc2.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		fnconstg 6716	. . 3 ⊢ (𝐵 ∈ 𝑊 → (𝐴 × {𝐵}) Fn 𝐴)
4	2, 3	syl 17	. 2 ⊢ (𝜑 → (𝐴 × {𝐵}) Fn 𝐴)
5		ofc2.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		inidm 4180	. 2 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7		ofc2.4	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
8		fvconst2g 7142	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
9	2, 8	sylan 580	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
10	1, 4, 5, 5, 6, 7, 9	ofval 7628	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {csn 4579   × cxp 5621   Fn wfn 6481  ‘cfv 6486  (class class class)co 7353   ∘_f cof 7615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-of 7617

This theorem is referenced by:  lflvscl  39075  lkrsc  39095  ldualvsval  39136