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Theorem ofrval 6805
 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 2611 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2611 . . . . . 6 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7ofrfval 6803 . . . . 5 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
98biimpa 500 . . . 4 ((𝜑𝐹𝑟 𝑅𝐺) → ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥))
10 fveq2 6103 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
11 fveq2 6103 . . . . . 6 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
1210, 11breq12d 4596 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ (𝐹𝑋)𝑅(𝐺𝑋)))
1312rspccv 3279 . . . 4 (∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥) → (𝑋𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
149, 13syl 17 . . 3 ((𝜑𝐹𝑟 𝑅𝐺) → (𝑋𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
15143impia 1253 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐹𝑋)𝑅(𝐺𝑋))
16 simp1 1054 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝜑)
17 inss1 3795 . . . . 5 (𝐴𝐵) ⊆ 𝐴
185, 17eqsstr3i 3599 . . . 4 𝑆𝐴
19 simp3 1056 . . . 4 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝑆)
2018, 19sseldi 3566 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝐴)
21 ofval.6 . . 3 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2216, 20, 21syl2anc 691 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐹𝑋) = 𝐶)
23 inss2 3796 . . . . 5 (𝐴𝐵) ⊆ 𝐵
245, 23eqsstr3i 3599 . . . 4 𝑆𝐵
2524, 19sseldi 3566 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝐵)
26 ofval.7 . . 3 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2716, 25, 26syl2anc 691 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐺𝑋) = 𝐷)
2815, 22, 273brtr3d 4614 1 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝐶𝑅𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∩ cin 3539   class class class wbr 4583   Fn wfn 5799  ‘cfv 5804   ∘𝑟 cofr 6794 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ofr 6796 This theorem is referenced by:  itg1le  23286  gsumle  29110  ftc1anclem5  32659
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