Theorem offveqb 7688

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 6940	. . . 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 4210	. . . 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 7672	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
11	3, 10	eqeq12d 2740	. 2 ⊢ (𝜑 → (𝐻 = (𝐹 ∘f 𝑅𝐺) ↔ (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))))
12		fvexd 6896	. . . 4 ⊢ (𝜑 → (𝐻‘𝑥) ∈ V)
13	12	ralrimivw 3142	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) ∈ V)
14		mpteqb 7007	. . 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 279	1 ⊢ (𝜑 → (𝐻 = (𝐹 ∘f 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3053  Vcvv 3466   ↦ cmpt 5221   Fn wfn 6528  ‘cfv 6533  (class class class)co 7401   ∘_f cof 7661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663

This theorem is referenced by:  eqlkr2  38426