Theorem ofrfval 5662

Description: Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
offval.1	⊢ (φ → 𝐹 Fn A)
offval.2	⊢ (φ → 𝐺 Fn B)
offval.3	⊢ (φ → A ∈ 𝑉)
offval.4	⊢ (φ → B ∈ 𝑊)
offval.5	⊢ (A ∩ B) = 𝑆
offval.6	⊢ ((φ ∧ x ∈ A) → (𝐹‘x) = 𝐶)
offval.7	⊢ ((φ ∧ x ∈ B) → (𝐺‘x) = 𝐷)

Assertion

Ref	Expression
ofrfval	⊢ (φ → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀x ∈ 𝑆 𝐶𝑅𝐷))

Distinct variable groups:   x,A   x,𝐹   x,𝐺   φ,x   x,𝑆   x,𝑅

Allowed substitution hints:   B(x)   𝐶(x)   𝐷(x)   𝑉(x)   𝑊(x)

Proof of Theorem ofrfval

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval.1	. . . 4 ⊢ (φ → 𝐹 Fn A)
2		offval.3	. . . 4 ⊢ (φ → A ∈ 𝑉)
3		fnex 5326	. . . 4 ⊢ ((𝐹 Fn A ∧ A ∈ 𝑉) → 𝐹 ∈ V)
4	1, 2, 3	syl2anc 391	. . 3 ⊢ (φ → 𝐹 ∈ V)
5		offval.2	. . . 4 ⊢ (φ → 𝐺 Fn B)
6		offval.4	. . . 4 ⊢ (φ → B ∈ 𝑊)
7		fnex 5326	. . . 4 ⊢ ((𝐺 Fn B ∧ B ∈ 𝑊) → 𝐺 ∈ V)
8	5, 6, 7	syl2anc 391	. . 3 ⊢ (φ → 𝐺 ∈ V)
9		dmeq 4478	. . . . . 6 ⊢ (f = 𝐹 → dom f = dom 𝐹)
10		dmeq 4478	. . . . . 6 ⊢ (g = 𝐺 → dom g = dom 𝐺)
11	9, 10	ineqan12d 3134	. . . . 5 ⊢ ((f = 𝐹 ∧ g = 𝐺) → (dom f ∩ dom g) = (dom 𝐹 ∩ dom 𝐺))
12		fveq1 5120	. . . . . 6 ⊢ (f = 𝐹 → (f‘x) = (𝐹‘x))
13		fveq1 5120	. . . . . 6 ⊢ (g = 𝐺 → (g‘x) = (𝐺‘x))
14	12, 13	breqan12d 3770	. . . . 5 ⊢ ((f = 𝐹 ∧ g = 𝐺) → ((f‘x)𝑅(g‘x) ↔ (𝐹‘x)𝑅(𝐺‘x)))
15	11, 14	raleqbidv 2511	. . . 4 ⊢ ((f = 𝐹 ∧ g = 𝐺) → (∀x ∈ (dom f ∩ dom g)(f‘x)𝑅(g‘x) ↔ ∀x ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘x)𝑅(𝐺‘x)))
16		df-ofr 5655	. . . 4 ⊢ ∘𝑟 𝑅 = {⟨f, g⟩ ∣ ∀x ∈ (dom f ∩ dom g)(f‘x)𝑅(g‘x)}
17	15, 16	brabga 3992	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀x ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘x)𝑅(𝐺‘x)))
18	4, 8, 17	syl2anc 391	. 2 ⊢ (φ → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀x ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘x)𝑅(𝐺‘x)))
19		fndm 4941	. . . . . 6 ⊢ (𝐹 Fn A → dom 𝐹 = A)
20	1, 19	syl 14	. . . . 5 ⊢ (φ → dom 𝐹 = A)
21		fndm 4941	. . . . . 6 ⊢ (𝐺 Fn B → dom 𝐺 = B)
22	5, 21	syl 14	. . . . 5 ⊢ (φ → dom 𝐺 = B)
23	20, 22	ineq12d 3133	. . . 4 ⊢ (φ → (dom 𝐹 ∩ dom 𝐺) = (A ∩ B))
24		offval.5	. . . 4 ⊢ (A ∩ B) = 𝑆
25	23, 24	syl6eq 2085	. . 3 ⊢ (φ → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
26	25	raleqdv 2505	. 2 ⊢ (φ → (∀x ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘x)𝑅(𝐺‘x) ↔ ∀x ∈ 𝑆 (𝐹‘x)𝑅(𝐺‘x)))
27		inss1 3151	. . . . . . 7 ⊢ (A ∩ B) ⊆ A
28	24, 27	eqsstr3i 2970	. . . . . 6 ⊢ 𝑆 ⊆ A
29	28	sseli 2935	. . . . 5 ⊢ (x ∈ 𝑆 → x ∈ A)
30		offval.6	. . . . 5 ⊢ ((φ ∧ x ∈ A) → (𝐹‘x) = 𝐶)
31	29, 30	sylan2 270	. . . 4 ⊢ ((φ ∧ x ∈ 𝑆) → (𝐹‘x) = 𝐶)
32		inss2 3152	. . . . . . 7 ⊢ (A ∩ B) ⊆ B
33	24, 32	eqsstr3i 2970	. . . . . 6 ⊢ 𝑆 ⊆ B
34	33	sseli 2935	. . . . 5 ⊢ (x ∈ 𝑆 → x ∈ B)
35		offval.7	. . . . 5 ⊢ ((φ ∧ x ∈ B) → (𝐺‘x) = 𝐷)
36	34, 35	sylan2 270	. . . 4 ⊢ ((φ ∧ x ∈ 𝑆) → (𝐺‘x) = 𝐷)
37	31, 36	breq12d 3768	. . 3 ⊢ ((φ ∧ x ∈ 𝑆) → ((𝐹‘x)𝑅(𝐺‘x) ↔ 𝐶𝑅𝐷))
38	37	ralbidva 2316	. 2 ⊢ (φ → (∀x ∈ 𝑆 (𝐹‘x)𝑅(𝐺‘x) ↔ ∀x ∈ 𝑆 𝐶𝑅𝐷))
39	18, 26, 38	3bitrd 203	1 ⊢ (φ → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀x ∈ 𝑆 𝐶𝑅𝐷))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Vcvv 2551   ∩ cin 2910   class class class wbr 3755  dom cdm 4288   Fn wfn 4840  ‘cfv 4845   ∘_𝑟 cofr 5653

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-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

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  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-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  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-ofr 5655

This theorem is referenced by:  ofrval  5664  ofrfval2  5669  caofref  5674  caofrss  5677  caoftrn  5678