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Theorem ofrval 5633
 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 2014 . . . . . 6 ((φ x A) → (𝐹x) = (𝐹x))
7 eqidd 2014 . . . . . 6 ((φ x B) → (𝐺x) = (𝐺x))
81, 2, 3, 4, 5, 6, 7ofrfval 5631 . . . . 5 (φ → (𝐹𝑟 𝑅𝐺x 𝑆 (𝐹x)𝑅(𝐺x)))
98biimpa 280 . . . 4 ((φ 𝐹𝑟 𝑅𝐺) → x 𝑆 (𝐹x)𝑅(𝐺x))
10 fveq2 5091 . . . . . 6 (x = 𝑋 → (𝐹x) = (𝐹𝑋))
11 fveq2 5091 . . . . . 6 (x = 𝑋 → (𝐺x) = (𝐺𝑋))
1210, 11breq12d 3740 . . . . 5 (x = 𝑋 → ((𝐹x)𝑅(𝐺x) ↔ (𝐹𝑋)𝑅(𝐺𝑋)))
1312rspccv 2621 . . . 4 (x 𝑆 (𝐹x)𝑅(𝐺x) → (𝑋 𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
149, 13syl 14 . . 3 ((φ 𝐹𝑟 𝑅𝐺) → (𝑋 𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
15143impia 1082 . 2 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → (𝐹𝑋)𝑅(𝐺𝑋))
16 simp1 886 . . 3 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → φ)
17 inss1 3125 . . . . 5 (AB) ⊆ A
185, 17eqsstr3i 2944 . . . 4 𝑆A
19 simp3 888 . . . 4 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → 𝑋 𝑆)
2018, 19sseldi 2911 . . 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 3126 . . . . 5 (AB) ⊆ B
245, 23eqsstr3i 2944 . . . 4 𝑆B
2524, 19sseldi 2911 . . 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 3756 1 ((φ 𝐹𝑟 𝑅𝐺 𝑋 𝑆) → 𝐶𝑅𝐷)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 867   = wceq 1223   ∈ wcel 1366  ∀wral 2275   ∩ cin 2884   class class class wbr 3727   Fn wfn 4812  ‘cfv 4817   ∘𝑟 cofr 5622 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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-coll 3835  ax-sep 3838  ax-pow 3890  ax-pr 3907 This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-rex 2281  df-reu 2282  df-rab 2284  df-v 2528  df-sbc 2733  df-csb 2821  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-uni 3544  df-iun 3622  df-br 3728  df-opab 3782  df-mpt 3783  df-id 3993  df-xp 4266  df-rel 4267  df-cnv 4268  df-co 4269  df-dm 4270  df-rn 4271  df-res 4272  df-ima 4273  df-iota 4782  df-fun 4819  df-fn 4820  df-f 4821  df-f1 4822  df-fo 4823  df-f1o 4824  df-fv 4825  df-ofr 5624 This theorem is referenced by: (None)
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