Theorem ofrval 7683

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
ofval.6	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
ofval.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 2736	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2736	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	ofrfval 7681	. . . . 5 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
9	8	biimpa 476	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥))
10		fveq2 6876	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
11		fveq2 6876	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
12	10, 11	breq12d 5132	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ (𝐹‘𝑋)𝑅(𝐺‘𝑋)))
13	12	rspccv 3598	. . . 4 ⊢ (∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋)𝑅(𝐺‘𝑋)))
14	9, 13	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋)𝑅(𝐺‘𝑋)))
15	14	3impia 1117	. 2 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋)𝑅(𝐺‘𝑋))
16		simp1 1136	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝜑)
17		inss1 4212	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
18	5, 17	eqsstrri 4006	. . . 4 ⊢ 𝑆 ⊆ 𝐴
19		simp3 1138	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
20	18, 19	sselid 3956	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐴)
21		ofval.6	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
22	16, 20, 21	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶)
23		inss2 4213	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
24	5, 23	eqsstrri 4006	. . . 4 ⊢ 𝑆 ⊆ 𝐵
25	24, 19	sselid 3956	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵)
26		ofval.7	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)
27	16, 25, 26	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷)
28	15, 22, 27	3brtr3d 5150	1 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051   ∩ cin 3925   class class class wbr 5119   Fn wfn 6526  ‘cfv 6531   ∘_r cofr 7670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ofr 7672

This theorem is referenced by:  mhpmulcl  22087  itg1le  25666  gsumle  33092  ftc1anclem5  37721