Theorem offveqb 7536

Description: Equivalent expressions for equality with a function operation. (Contributed by NM, 9-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)

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)

Assertion

Ref	Expression
offveqb	⊢ (𝜑 → (𝐻 = (𝐹 ∘f 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶)))

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 offveqb

Step	Hyp	Ref	Expression
1		offveq.4	. . . 4 ⊢ (𝜑 → 𝐻 Fn 𝐴)
2		dffn5 6810	. . . 4 ⊢ (𝐻 Fn 𝐴 ↔ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)))
3	1, 2	sylib 217	. . 3 ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)))
4		offveq.2	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
5		offveq.3	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐴)
6		offveq.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
7		inidm 4149	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 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, 6, 7, 8, 9	offval 7520	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
11	3, 10	eqeq12d 2754	. 2 ⊢ (𝜑 → (𝐻 = (𝐹 ∘f 𝑅𝐺) ↔ (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))))
12		fvexd 6771	. . . 4 ⊢ (𝜑 → (𝐻‘𝑥) ∈ V)
13	12	ralrimivw 3108	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) ∈ V)
14		mpteqb 6876	. . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝐻‘𝑥) ∈ V → ((𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶)))
15	13, 14	syl 17	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶)))
16	11, 15	bitrd 278	1 ⊢ (𝜑 → (𝐻 = (𝐹 ∘f 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ↦ cmpt 5153   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255   ∘_f cof 7509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511

This theorem is referenced by:  eqlkr2  37041