Theorem ofc2 6921

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 6093	. . 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 3822	. 2 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7		ofc2.4	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
8		fvconst2g 6467	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
9	2, 8	sylan 488	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐴 × {𝐵})‘𝑋) = 𝐵)
10	1, 4, 5, 5, 6, 7, 9	ofval 6906	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅(𝐴 × {𝐵}))‘𝑋) = (𝐶𝑅𝐵))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {csn 4177   × cxp 5112   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by:  lflvscl  34364  lkrsc  34384  ldualvsval  34425