Theorem offveqb 5649

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 5140	. . . 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 3119	. . . 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 5638	. . 3 ⊢ (φ → (𝐹 ∘𝑓 𝑅𝐺) = (x ∈ A ↦ (B𝑅𝐶)))
11	3, 10	eqeq12d 2032	. 2 ⊢ (φ → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ (x ∈ A ↦ (𝐻‘x)) = (x ∈ A ↦ (B𝑅𝐶))))
12		funfvex 5113	. . . . . 6 ⊢ ((Fun 𝐻 ∧ x ∈ dom 𝐻) → (𝐻‘x) ∈ V)
13	12	funfni 4921	. . . . 5 ⊢ ((𝐻 Fn A ∧ x ∈ A) → (𝐻‘x) ∈ V)
14	1, 13	sylan 267	. . . 4 ⊢ ((φ ∧ x ∈ A) → (𝐻‘x) ∈ V)
15	14	ralrimiva 2366	. . 3 ⊢ (φ → ∀x ∈ A (𝐻‘x) ∈ V)
16		mpteqb 5182	. . 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 1226   ∈ wcel 1370  ∀wral 2280  Vcvv 2531   ↦ cmpt 3788   Fn wfn 4820  ‘cfv 4825  (class class class)co 5432   ∘_𝑓 cof 5629

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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-coll 3842  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-setind 4200

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ne 2184  df-ral 2285  df-rex 2286  df-reu 2287  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-dif 2893  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-ov 5435  df-oprab 5436  df-mpt2 5437  df-of 5631

This theorem is referenced by: (None)