Theorem offveqb 6919

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 6241	. . . 4 ⊢ (𝐻 Fn 𝐴 ↔ 𝐻 = (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)))
3	1, 2	sylib 208	. . 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 3822	. . . 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 6904	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶)))
11	3, 10	eqeq12d 2637	. 2 ⊢ (𝜑 → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ (𝑥 ∈ 𝐴 ↦ (𝐻‘𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵𝑅𝐶))))
12		fvexd 6203	. . . 4 ⊢ (𝜑 → (𝐻‘𝑥) ∈ V)
13	12	ralrimivw 2967	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) ∈ V)
14		mpteqb 6299	. . 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 268	1 ⊢ (𝜑 → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ↦ cmpt 4729   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-8 1992  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-pow 4843  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:  eqlkr2  34387