Theorem ofrval 5633

Description: Exhibit a function relation at a point. (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
ofrval.6	⊢ ((φ ∧ 𝑋 ∈ A) → (𝐹‘𝑋) = 𝐶)
ofrval.7	⊢ ((φ ∧ 𝑋 ∈ B) → (𝐺‘𝑋) = 𝐷)

Assertion

Ref	Expression
ofrval	⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷)

Proof of Theorem ofrval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval.1	. . . . . 6 ⊢ (φ → 𝐹 Fn A)
2		offval.2	. . . . . 6 ⊢ (φ → 𝐺 Fn B)
3		offval.3	. . . . . 6 ⊢ (φ → A ∈ 𝑉)
4		offval.4	. . . . . 6 ⊢ (φ → B ∈ 𝑊)
5		offval.5	. . . . . 6 ⊢ (A ∩ B) = 𝑆
6		eqidd 2014	. . . . . 6 ⊢ ((φ ∧ x ∈ A) → (𝐹‘x) = (𝐹‘x))
7		eqidd 2014	. . . . . 6 ⊢ ((φ ∧ x ∈ B) → (𝐺‘x) = (𝐺‘x))
8	1, 2, 3, 4, 5, 6, 7	ofrfval 5631	. . . . 5 ⊢ (φ → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀x ∈ 𝑆 (𝐹‘x)𝑅(𝐺‘x)))
9	8	biimpa 280	. . . 4 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺) → ∀x ∈ 𝑆 (𝐹‘x)𝑅(𝐺‘x))
10		fveq2 5091	. . . . . 6 ⊢ (x = 𝑋 → (𝐹‘x) = (𝐹‘𝑋))
11		fveq2 5091	. . . . . 6 ⊢ (x = 𝑋 → (𝐺‘x) = (𝐺‘𝑋))
12	10, 11	breq12d 3740	. . . . 5 ⊢ (x = 𝑋 → ((𝐹‘x)𝑅(𝐺‘x) ↔ (𝐹‘𝑋)𝑅(𝐺‘𝑋)))
13	12	rspccv 2621	. . . 4 ⊢ (∀x ∈ 𝑆 (𝐹‘x)𝑅(𝐺‘x) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋)𝑅(𝐺‘𝑋)))
14	9, 13	syl 14	. . 3 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋)𝑅(𝐺‘𝑋)))
15	14	3impia 1082	. 2 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋)𝑅(𝐺‘𝑋))
16		simp1 886	. . 3 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → φ)
17		inss1 3125	. . . . 5 ⊢ (A ∩ B) ⊆ A
18	5, 17	eqsstr3i 2944	. . . 4 ⊢ 𝑆 ⊆ A
19		simp3 888	. . . 4 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
20	18, 19	sseldi 2911	. . 3 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ A)
21		ofrval.6	. . 3 ⊢ ((φ ∧ 𝑋 ∈ A) → (𝐹‘𝑋) = 𝐶)
22	16, 20, 21	syl2anc 391	. 2 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶)
23		inss2 3126	. . . . 5 ⊢ (A ∩ B) ⊆ B
24	5, 23	eqsstr3i 2944	. . . 4 ⊢ 𝑆 ⊆ B
25	24, 19	sseldi 2911	. . 3 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ B)
26		ofrval.7	. . 3 ⊢ ((φ ∧ 𝑋 ∈ B) → (𝐺‘𝑋) = 𝐷)
27	16, 25, 26	syl2anc 391	. 2 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷)
28	15, 22, 27	3brtr3d 3756	1 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 867   = wceq 1223   ∈ wcel 1366  ∀wral 2275   ∩ cin 2884   class class class wbr 3727   Fn wfn 4812  ‘cfv 4817   ∘_𝑟 cofr 5622

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 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-coll 3835  ax-sep 3838  ax-pow 3890  ax-pr 3907

This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-reu 2282  df-rab 2284  df-v 2528  df-sbc 2733  df-csb 2821  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-iun 3622  df-br 3728  df-opab 3782  df-mpt 3783  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-f 4821  df-f1 4822  df-fo 4823  df-f1o 4824  df-fv 4825  df-ofr 5624

This theorem is referenced by: (None)