Theorem offveq 7652

Description: Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
offveq.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offveq.2	⊢ (𝜑 → 𝐹 Fn 𝐴)
offveq.3	⊢ (𝜑 → 𝐺 Fn 𝐴)
offveq.4	⊢ (𝜑 → 𝐻 Fn 𝐴)
offveq.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
offveq.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)
offveq.7	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥))

Assertion

Ref	Expression
offveq	⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = 𝐻)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝑥,𝐻   𝜑,𝑥   𝑥,𝑅

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem offveq

Step	Hyp	Ref	Expression
1		offveq.2	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		offveq.3	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
3		offveq.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		inidm 4168	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
5	1, 2, 3, 3, 4	offn 7639	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝐴)
6		offveq.4	. 2 ⊢ (𝜑 → 𝐻 Fn 𝐴)
7		offveq.5	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
8		offveq.6	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)
9	1, 2, 3, 3, 4, 7, 8	ofval 7637	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶))
10		offveq.7	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥))
11	9, 10	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐻‘𝑥))
12	5, 6, 11	eqfnfvd 6982	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = 𝐻)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114   Fn wfn 6489  ‘cfv 6494  (class class class)co 7362   ∘_f cof 7624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7626

This theorem is referenced by:  caofid0l  7659  caofid0r  7660  caofid1  7661  caofid2  7662  ofnegsub  12152  psdmul  22146  bddibl  25821  dvaddf  25923  plydivlem3  26276  psrmonprod  33715  poimirlem5  37964  poimirlem10  37969  poimirlem22  37981  fsuppssind  43044  ofsubid  44773  ofmul12  44774  ofdivrec  44775  ofdivcan4  44776  ofdivdiv2  44777