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Theorem ofrval 5999
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 (𝜑𝐴𝑉)
offval.4 (𝜑𝐵𝑊)
offval.5 (𝐴𝐵) = 𝑆
ofrval.6 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
ofrval.7 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
Assertion
Ref Expression
ofrval ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 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 (𝜑𝐴𝑉)
4 offval.4 . . . . . 6 (𝜑𝐵𝑊)
5 offval.5 . . . . . 6 (𝐴𝐵) = 𝑆
6 eqidd 2141 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2141 . . . . . 6 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7ofrfval 5997 . . . . 5 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
98biimpa 294 . . . 4 ((𝜑𝐹𝑟 𝑅𝐺) → ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥))
10 fveq2 5428 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
11 fveq2 5428 . . . . . 6 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
1210, 11breq12d 3949 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ (𝐹𝑋)𝑅(𝐺𝑋)))
1312rspccv 2789 . . . 4 (∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥) → (𝑋𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
149, 13syl 14 . . 3 ((𝜑𝐹𝑟 𝑅𝐺) → (𝑋𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
15143impia 1179 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐹𝑋)𝑅(𝐺𝑋))
16 simp1 982 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝜑)
17 inss1 3300 . . . . 5 (𝐴𝐵) ⊆ 𝐴
185, 17eqsstrri 3134 . . . 4 𝑆𝐴
19 simp3 984 . . . 4 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝑆)
2018, 19sseldi 3099 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝐴)
21 ofrval.6 . . 3 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2216, 20, 21syl2anc 409 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐹𝑋) = 𝐶)
23 inss2 3301 . . . . 5 (𝐴𝐵) ⊆ 𝐵
245, 23eqsstrri 3134 . . . 4 𝑆𝐵
2524, 19sseldi 3099 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝐵)
26 ofrval.7 . . 3 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2716, 25, 26syl2anc 409 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐺𝑋) = 𝐷)
2815, 22, 273brtr3d 3966 1 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝐶𝑅𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 963   = wceq 1332  wcel 1481  wral 2417  cin 3074   class class class wbr 3936   Fn wfn 5125  cfv 5130  𝑟 cofr 5988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ofr 5990
This theorem is referenced by: (None)
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