Theorem ofrfval 5634

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 5299	. . . 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 5299	. . . 4 ⊢ ((𝐺 Fn B ∧ B ∈ 𝑊) → 𝐺 ∈ V)
8	5, 6, 7	syl2anc 391	. . 3 ⊢ (φ → 𝐺 ∈ V)
9		dmeq 4453	. . . . . 6 ⊢ (f = 𝐹 → dom f = dom 𝐹)
10		dmeq 4453	. . . . . 6 ⊢ (g = 𝐺 → dom g = dom 𝐺)
11	9, 10	ineqan12d 3111	. . . . 5 ⊢ ((f = 𝐹 ∧ g = 𝐺) → (dom f ∩ dom g) = (dom 𝐹 ∩ dom 𝐺))
12		fveq1 5093	. . . . . 6 ⊢ (f = 𝐹 → (f‘x) = (𝐹‘x))
13		fveq1 5093	. . . . . 6 ⊢ (g = 𝐺 → (g‘x) = (𝐺‘x))
14	12, 13	breqan12d 3745	. . . . 5 ⊢ ((f = 𝐹 ∧ g = 𝐺) → ((f‘x)𝑅(g‘x) ↔ (𝐹‘x)𝑅(𝐺‘x)))
15	11, 14	raleqbidv 2489	. . . 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 5627	. . . 4 ⊢ ∘𝑟 𝑅 = {⟨f, g⟩ ∣ ∀x ∈ (dom f ∩ dom g)(f‘x)𝑅(g‘x)}
17	15, 16	brabga 3967	. . 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 4915	. . . . . 6 ⊢ (𝐹 Fn A → dom 𝐹 = A)
20	1, 19	syl 14	. . . . 5 ⊢ (φ → dom 𝐹 = A)
21		fndm 4915	. . . . . 6 ⊢ (𝐺 Fn B → dom 𝐺 = B)
22	5, 21	syl 14	. . . . 5 ⊢ (φ → dom 𝐺 = B)
23	20, 22	ineq12d 3110	. . . 4 ⊢ (φ → (dom 𝐹 ∩ dom 𝐺) = (A ∩ B))
24		offval.5	. . . 4 ⊢ (A ∩ B) = 𝑆
25	23, 24	syl6eq 2064	. . 3 ⊢ (φ → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
26	25	raleqdv 2483	. 2 ⊢ (φ → (∀x ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘x)𝑅(𝐺‘x) ↔ ∀x ∈ 𝑆 (𝐹‘x)𝑅(𝐺‘x)))
27		inss1 3128	. . . . . . 7 ⊢ (A ∩ B) ⊆ A
28	24, 27	eqsstr3i 2947	. . . . . 6 ⊢ 𝑆 ⊆ A
29	28	sseli 2912	. . . . 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 3129	. . . . . . 7 ⊢ (A ∩ B) ⊆ B
33	24, 32	eqsstr3i 2947	. . . . . 6 ⊢ 𝑆 ⊆ B
34	33	sseli 2912	. . . . 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 3743	. . 3 ⊢ ((φ ∧ x ∈ 𝑆) → ((𝐹‘x)𝑅(𝐺‘x) ↔ 𝐶𝑅𝐷))
38	37	ralbidva 2294	. 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 1226   ∈ wcel 1369  ∀wral 2278  Vcvv 2529   ∩ cin 2887   class class class wbr 3730  dom cdm 4263   Fn wfn 4815  ‘cfv 4820   ∘_𝑟 cofr 5625

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 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-14 1381  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998  ax-coll 3838  ax-sep 3841  ax-pow 3893  ax-pr 3910

This theorem depends on definitions:  df-bi 110  df-3an 871  df-tru 1229  df-nf 1326  df-sb 1622  df-eu 1879  df-mo 1880  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-ral 2283  df-rex 2284  df-reu 2285  df-rab 2287  df-v 2531  df-sbc 2736  df-csb 2824  df-un 2893  df-in 2895  df-ss 2902  df-pw 3328  df-sn 3348  df-pr 3349  df-op 3351  df-uni 3547  df-iun 3625  df-br 3731  df-opab 3785  df-mpt 3786  df-id 3996  df-xp 4269  df-rel 4270  df-cnv 4271  df-co 4272  df-dm 4273  df-rn 4274  df-res 4275  df-ima 4276  df-iota 4785  df-fun 4822  df-fn 4823  df-f 4824  df-f1 4825  df-fo 4826  df-f1o 4827  df-fv 4828  df-ofr 5627

This theorem is referenced by:  ofrval  5636  ofrfval2  5641  caofref  5646  caofrss  5649  caoftrn  5650