Theorem ofrfval 6144

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 5784	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
4	1, 2, 3	syl2anc 411	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
5		offval.2	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
6		offval.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		fnex 5784	. . . 4 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ 𝑊) → 𝐺 ∈ V)
8	5, 6, 7	syl2anc 411	. . 3 ⊢ (𝜑 → 𝐺 ∈ V)
9		dmeq 4866	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
10		dmeq 4866	. . . . . 6 ⊢ (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
11	9, 10	ineqan12d 3366	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
12		fveq1 5557	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
13		fveq1 5557	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥))
14	12, 13	breqan12d 4049	. . . . 5 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)𝑅(𝑔‘𝑥) ↔ (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
15	11, 14	raleqbidv 2709	. . . 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 6136	. . . 4 ⊢ ∘𝑟 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓‘𝑥)𝑅(𝑔‘𝑥)}
17	15, 16	brabga 4298	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥)))
18	4, 8, 17	syl2anc 411	. 2 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥)))
19		fndm 5357	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
20	1, 19	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
21		fndm 5357	. . . . . 6 ⊢ (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
22	5, 21	syl 14	. . . . 5 ⊢ (𝜑 → dom 𝐺 = 𝐵)
23	20, 22	ineq12d 3365	. . . 4 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴 ∩ 𝐵))
24		offval.5	. . . 4 ⊢ (𝐴 ∩ 𝐵) = 𝑆
25	23, 24	eqtrdi 2245	. . 3 ⊢ (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
26	25	raleqdv 2699	. 2 ⊢ (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
27		inss1 3383	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
28	24, 27	eqsstrri 3216	. . . . . 6 ⊢ 𝑆 ⊆ 𝐴
29	28	sseli 3179	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐴)
30		offval.6	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
31	29, 30	sylan2 286	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐹‘𝑥) = 𝐶)
32		inss2 3384	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
33	24, 32	eqsstrri 3216	. . . . . 6 ⊢ 𝑆 ⊆ 𝐵
34	33	sseli 3179	. . . . 5 ⊢ (𝑥 ∈ 𝑆 → 𝑥 ∈ 𝐵)
35		offval.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)
36	34, 35	sylan2 286	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐺‘𝑥) = 𝐷)
37	31, 36	breq12d 4046	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ 𝐶𝑅𝐷))
38	37	ralbidva 2493	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))
39	18, 26, 38	3bitrd 214	1 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763   ∩ cin 3156   class class class wbr 4033  dom cdm 4663   Fn wfn 5253  ‘cfv 5258   ∘_𝑟 cofr 6134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ofr 6136

This theorem is referenced by:  ofrval  6146  ofrfval2  6152  caofref  6159  caofrss  6162  caoftrn  6163