Theorem ofrval 6277

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 2233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
7		eqidd 2233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝑥))
8	1, 2, 3, 4, 5, 6, 7	ofrfval 6275	. . . . 5 ⊢ (𝜑 → (𝐹 ∘𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥)))
9	8	biimpa 296	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺) → ∀𝑥 ∈ 𝑆 (𝐹‘𝑥)𝑅(𝐺‘𝑥))
10		fveq2 5670	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
11		fveq2 5670	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐺‘𝑥) = (𝐺‘𝑋))
12	10, 11	breq12d 4122	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥)𝑅(𝐺‘𝑥) ↔ (𝐹‘𝑋)𝑅(𝐺‘𝑋)))
13	12	rspccv 2918	. . . 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 1227	. 2 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋)𝑅(𝐺‘𝑋))
16		simp1 1024	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝜑)
17		inss1 3441	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
18	5, 17	eqsstrri 3271	. . . 4 ⊢ 𝑆 ⊆ 𝐴
19		simp3 1026	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
20	18, 19	sselid 3236	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐴)
21		ofrval.6	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐶)
22	16, 20, 21	syl2anc 411	. 2 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋) = 𝐶)
23		inss2 3442	. . . . 5 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
24	5, 23	eqsstrri 3271	. . . 4 ⊢ 𝑆 ⊆ 𝐵
25	24, 19	sselid 3236	. . 3 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝐵)
26		ofrval.7	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐺‘𝑋) = 𝐷)
27	16, 25, 26	syl2anc 411	. 2 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐺‘𝑋) = 𝐷)
28	15, 22, 27	3brtr3d 4140	1 ⊢ ((𝜑 ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  ∀wral 2520   ∩ cin 3210   class class class wbr 4109   Fn wfn 5347  ‘cfv 5352   ∘_𝑟 cofr 6265

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ofr 6267

This theorem is referenced by:  psrbaglesuppg  14821