Theorem offveq 6317

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.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥))
2	1	eqcomd 2244	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = (𝐵𝑅𝐶))
3	2	ralrimiva 2623	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶))
4		offveq.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		offveq.2	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
6		offveq.3	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐴)
7		offveq.4	. . . 4 ⊢ (𝜑 → 𝐻 Fn 𝐴)
8		offveq.5	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
9		offveq.6	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)
10	4, 5, 6, 7, 8, 9	offveqb 6316	. . 3 ⊢ (𝜑 → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶)))
11	3, 10	mpbird 167	. 2 ⊢ (𝜑 → 𝐻 = (𝐹 ∘𝑓 𝑅𝐺))
12	11	eqcomd 2244	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528   Fn wfn 5370  ‘cfv 5375  (class class class)co 6079   ∘_𝑓 cof 6294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-setind 4682

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-ov 6082  df-oprab 6083  df-mpo 6084  df-of 6296

This theorem is referenced by:  caofid0l  6323  caofid0r  6324  caofid1  6325  caofid2  6326