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Theorem ofrval 5664
Description: Exhibit a function relation at a point. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
offval.1 (φ𝐹 Fn A)
offval.2 (φ𝐺 Fn B)
offval.3 (φA 𝑉)
offval.4 (φB 𝑊)
offval.5 (AB) = 𝑆
ofrval.6 ((φ 𝑋 A) → (𝐹𝑋) = 𝐶)
ofrval.7 ((φ 𝑋 B) → (𝐺𝑋) = 𝐷)
Assertion
Ref Expression
ofrval ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → 𝐶𝑅𝐷)

Proof of Theorem ofrval
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . . . 6 (φ𝐹 Fn A)
2 offval.2 . . . . . 6 (φ𝐺 Fn B)
3 offval.3 . . . . . 6 (φA 𝑉)
4 offval.4 . . . . . 6 (φB 𝑊)
5 offval.5 . . . . . 6 (AB) = 𝑆
6 eqidd 2038 . . . . . 6 ((φ x A) → (𝐹x) = (𝐹x))
7 eqidd 2038 . . . . . 6 ((φ x B) → (𝐺x) = (𝐺x))
81, 2, 3, 4, 5, 6, 7ofrfval 5662 . . . . 5 (φ → (𝐹𝑟 𝑅𝐺x 𝑆 (𝐹x)𝑅(𝐺x)))
98biimpa 280 . . . 4 ((φ 𝐹𝑟 𝑅𝐺) → x 𝑆 (𝐹x)𝑅(𝐺x))
10 fveq2 5121 . . . . . 6 (x = 𝑋 → (𝐹x) = (𝐹𝑋))
11 fveq2 5121 . . . . . 6 (x = 𝑋 → (𝐺x) = (𝐺𝑋))
1210, 11breq12d 3768 . . . . 5 (x = 𝑋 → ((𝐹x)𝑅(𝐺x) ↔ (𝐹𝑋)𝑅(𝐺𝑋)))
1312rspccv 2647 . . . 4 (x 𝑆 (𝐹x)𝑅(𝐺x) → (𝑋 𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
149, 13syl 14 . . 3 ((φ 𝐹𝑟 𝑅𝐺) → (𝑋 𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
15143impia 1100 . 2 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → (𝐹𝑋)𝑅(𝐺𝑋))
16 simp1 903 . . 3 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → φ)
17 inss1 3151 . . . . 5 (AB) ⊆ A
185, 17eqsstr3i 2970 . . . 4 𝑆A
19 simp3 905 . . . 4 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → 𝑋 𝑆)
2018, 19sseldi 2937 . . 3 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → 𝑋 A)
21 ofrval.6 . . 3 ((φ 𝑋 A) → (𝐹𝑋) = 𝐶)
2216, 20, 21syl2anc 391 . 2 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → (𝐹𝑋) = 𝐶)
23 inss2 3152 . . . . 5 (AB) ⊆ B
245, 23eqsstr3i 2970 . . . 4 𝑆B
2524, 19sseldi 2937 . . 3 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → 𝑋 B)
26 ofrval.7 . . 3 ((φ 𝑋 B) → (𝐺𝑋) = 𝐷)
2716, 25, 26syl2anc 391 . 2 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → (𝐺𝑋) = 𝐷)
2815, 22, 273brtr3d 3784 1 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → 𝐶𝑅𝐷)
Colors of variables: wff set class
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-bnd 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)
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