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Theorem ofrval 6904
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 2622 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2622 . . . . . 6 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7ofrfval 6902 . . . . 5 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
98biimpa 501 . . . 4 ((𝜑𝐹𝑟 𝑅𝐺) → ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥))
10 fveq2 6189 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
11 fveq2 6189 . . . . . 6 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
1210, 11breq12d 4664 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ (𝐹𝑋)𝑅(𝐺𝑋)))
1312rspccv 3304 . . . 4 (∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥) → (𝑋𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
149, 13syl 17 . . 3 ((𝜑𝐹𝑟 𝑅𝐺) → (𝑋𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
15143impia 1260 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐹𝑋)𝑅(𝐺𝑋))
16 simp1 1060 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝜑)
17 inss1 3831 . . . . 5 (𝐴𝐵) ⊆ 𝐴
185, 17eqsstr3i 3634 . . . 4 𝑆𝐴
19 simp3 1062 . . . 4 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝑆)
2018, 19sseldi 3599 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝐴)
21 ofval.6 . . 3 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2216, 20, 21syl2anc 693 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐹𝑋) = 𝐶)
23 inss2 3832 . . . . 5 (𝐴𝐵) ⊆ 𝐵
245, 23eqsstr3i 3634 . . . 4 𝑆𝐵
2524, 19sseldi 3599 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝐵)
26 ofval.7 . . 3 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2716, 25, 26syl2anc 693 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐺𝑋) = 𝐷)
2815, 22, 273brtr3d 4682 1 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝐶𝑅𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  wral 2911  cin 3571   class class class wbr 4651   Fn wfn 5881  cfv 5886  𝑟 cofr 6893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ofr 6895
This theorem is referenced by:  itg1le  23474  gsumle  29764  ftc1anclem5  33469
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