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Theorem ofrval 7545
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
ofval.6 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
ofval.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 2739 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2739 . . . . . 6 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7ofrfval 7543 . . . . 5 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
98biimpa 477 . . . 4 ((𝜑𝐹r 𝑅𝐺) → ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥))
10 fveq2 6774 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
11 fveq2 6774 . . . . . 6 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
1210, 11breq12d 5087 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ (𝐹𝑋)𝑅(𝐺𝑋)))
1312rspccv 3558 . . . 4 (∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥) → (𝑋𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
149, 13syl 17 . . 3 ((𝜑𝐹r 𝑅𝐺) → (𝑋𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
15143impia 1116 . 2 ((𝜑𝐹r 𝑅𝐺𝑋𝑆) → (𝐹𝑋)𝑅(𝐺𝑋))
16 simp1 1135 . . 3 ((𝜑𝐹r 𝑅𝐺𝑋𝑆) → 𝜑)
17 inss1 4162 . . . . 5 (𝐴𝐵) ⊆ 𝐴
185, 17eqsstrri 3956 . . . 4 𝑆𝐴
19 simp3 1137 . . . 4 ((𝜑𝐹r 𝑅𝐺𝑋𝑆) → 𝑋𝑆)
2018, 19sselid 3919 . . 3 ((𝜑𝐹r 𝑅𝐺𝑋𝑆) → 𝑋𝐴)
21 ofval.6 . . 3 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2216, 20, 21syl2anc 584 . 2 ((𝜑𝐹r 𝑅𝐺𝑋𝑆) → (𝐹𝑋) = 𝐶)
23 inss2 4163 . . . . 5 (𝐴𝐵) ⊆ 𝐵
245, 23eqsstrri 3956 . . . 4 𝑆𝐵
2524, 19sselid 3919 . . 3 ((𝜑𝐹r 𝑅𝐺𝑋𝑆) → 𝑋𝐵)
26 ofval.7 . . 3 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2716, 25, 26syl2anc 584 . 2 ((𝜑𝐹r 𝑅𝐺𝑋𝑆) → (𝐺𝑋) = 𝐷)
2815, 22, 273brtr3d 5105 1 ((𝜑𝐹r 𝑅𝐺𝑋𝑆) → 𝐶𝑅𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  cin 3886   class class class wbr 5074   Fn wfn 6428  cfv 6433  r cofr 7532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ofr 7534
This theorem is referenced by:  mhpmulcl  21339  itg1le  24878  gsumle  31350  ftc1anclem5  35854
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