Theorem offveqb 5672

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		dffn5im 5162	. . . 4 ⊢ (𝐻 Fn A → 𝐻 = (x ∈ A ↦ (𝐻‘x)))
3	1, 2	syl 14	. . 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 3140	. . . 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 5661	. . 3 ⊢ (φ → (𝐹 ∘𝑓 𝑅𝐺) = (x ∈ A ↦ (B𝑅𝐶)))
11	3, 10	eqeq12d 2051	. 2 ⊢ (φ → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ (x ∈ A ↦ (𝐻‘x)) = (x ∈ A ↦ (B𝑅𝐶))))
12		funfvex 5135	. . . . . 6 ⊢ ((Fun 𝐻 ∧ x ∈ dom 𝐻) → (𝐻‘x) ∈ V)
13	12	funfni 4942	. . . . 5 ⊢ ((𝐻 Fn A ∧ x ∈ A) → (𝐻‘x) ∈ V)
14	1, 13	sylan 267	. . . 4 ⊢ ((φ ∧ x ∈ A) → (𝐻‘x) ∈ V)
15	14	ralrimiva 2386	. . 3 ⊢ (φ → ∀x ∈ A (𝐻‘x) ∈ V)
16		mpteqb 5204	. . 3 ⊢ (∀x ∈ A (𝐻‘x) ∈ V → ((x ∈ A ↦ (𝐻‘x)) = (x ∈ A ↦ (B𝑅𝐶)) ↔ ∀x ∈ A (𝐻‘x) = (B𝑅𝐶)))
17	15, 16	syl 14	. 2 ⊢ (φ → ((x ∈ A ↦ (𝐻‘x)) = (x ∈ A ↦ (B𝑅𝐶)) ↔ ∀x ∈ A (𝐻‘x) = (B𝑅𝐶)))
18	11, 17	bitrd 177	1 ⊢ (φ → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ ∀x ∈ A (𝐻‘x) = (B𝑅𝐶)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Vcvv 2551   ↦ cmpt 3809   Fn wfn 4840  ‘cfv 4845  (class class class)co 5455   ∘_𝑓 cof 5652

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-setind 4220

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-of 5654

This theorem is referenced by: (None)