Theorem ofrval 5664

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 2038	. . . . . 6 ⊢ ((φ ∧ x ∈ A) → (𝐹‘x) = (𝐹‘x))
7		eqidd 2038	. . . . . 6 ⊢ ((φ ∧ x ∈ B) → (𝐺‘x) = (𝐺‘x))
8	1, 2, 3, 4, 5, 6, 7	ofrfval 5662	. . . . 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 5121	. . . . . 6 ⊢ (x = 𝑋 → (𝐹‘x) = (𝐹‘𝑋))
11		fveq2 5121	. . . . . 6 ⊢ (x = 𝑋 → (𝐺‘x) = (𝐺‘𝑋))
12	10, 11	breq12d 3768	. . . . 5 ⊢ (x = 𝑋 → ((𝐹‘x)𝑅(𝐺‘x) ↔ (𝐹‘𝑋)𝑅(𝐺‘𝑋)))
13	12	rspccv 2647	. . . 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 1100	. 2 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → (𝐹‘𝑋)𝑅(𝐺‘𝑋))
16		simp1 903	. . 3 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → φ)
17		inss1 3151	. . . . 5 ⊢ (A ∩ B) ⊆ A
18	5, 17	eqsstr3i 2970	. . . 4 ⊢ 𝑆 ⊆ A
19		simp3 905	. . . 4 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝑋 ∈ 𝑆)
20	18, 19	sseldi 2937	. . 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 3152	. . . . 5 ⊢ (A ∩ B) ⊆ B
24	5, 23	eqsstr3i 2970	. . . 4 ⊢ 𝑆 ⊆ B
25	24, 19	sseldi 2937	. . 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 3784	1 ⊢ ((φ ∧ 𝐹 ∘𝑟 𝑅𝐺 ∧ 𝑋 ∈ 𝑆) → 𝐶𝑅𝐷)

Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ∩ cin 2910   class class class wbr 3755   Fn wfn 4840  ‘cfv 4845   ∘_𝑟 cofr 5653

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ofr 5655

This theorem is referenced by: (None)