Theorem ofrval 7672

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 2763	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2763	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	ofrfval 7670	. . . . 5 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
9	8	biimpa 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥))
10		fveq2 6867	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
11		fveq2 6867	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
12	10, 11	breq12d 5113	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ (𝐹‘𝑋)𝑅(𝐺‘𝑋)))
13	12	rspccv 3578	. . . 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 1130	. 2 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋)𝑅(𝐺‘𝑋))
16		simp1 1149	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝜑)
17		inss1 4188	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
18	5, 17	eqsstrri 3983	. . . 4 ⊢ 𝑆 ⊆ 𝐴
19		simp3 1151	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
20	18, 19	sselid 3934	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐴)
21		ofval.6	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
22	16, 20, 21	syl2anc 593	. 2 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶)
23		inss2 4189	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
24	5, 23	eqsstrri 3983	. . . 4 ⊢ 𝑆 ⊆ 𝐵
25	24, 19	sselid 3934	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵)
26		ofval.7	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)
27	16, 25, 26	syl2anc 593	. 2 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷)
28	15, 22, 27	3brtr3d 5131	1 ⊢ ((𝜑 ∧ 𝐹 ∘r 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1098   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ∩ cin 3903   class class class wbr 5100   Fn wfn 6516  ‘cfv 6521   ∘_r cofr 7659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ofr 7661

This theorem is referenced by:  gsumle  20185  mhpmulcl  22214  itg1le  25775  selvply1rhmlemb  33816  ftc1anclem5  38196