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Theorem ofrval 6118
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 2190 . . . . . 6 ((𝜑𝑥𝐴) → (𝐹𝑥) = (𝐹𝑥))
7 eqidd 2190 . . . . . 6 ((𝜑𝑥𝐵) → (𝐺𝑥) = (𝐺𝑥))
81, 2, 3, 4, 5, 6, 7ofrfval 6116 . . . . 5 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
98biimpa 296 . . . 4 ((𝜑𝐹𝑟 𝑅𝐺) → ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥))
10 fveq2 5534 . . . . . 6 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
11 fveq2 5534 . . . . . 6 (𝑥 = 𝑋 → (𝐺𝑥) = (𝐺𝑋))
1210, 11breq12d 4031 . . . . 5 (𝑥 = 𝑋 → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ (𝐹𝑋)𝑅(𝐺𝑋)))
1312rspccv 2853 . . . 4 (∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥) → (𝑋𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
149, 13syl 14 . . 3 ((𝜑𝐹𝑟 𝑅𝐺) → (𝑋𝑆 → (𝐹𝑋)𝑅(𝐺𝑋)))
15143impia 1202 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐹𝑋)𝑅(𝐺𝑋))
16 simp1 999 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝜑)
17 inss1 3370 . . . . 5 (𝐴𝐵) ⊆ 𝐴
185, 17eqsstrri 3203 . . . 4 𝑆𝐴
19 simp3 1001 . . . 4 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝑆)
2018, 19sselid 3168 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝐴)
21 ofrval.6 . . 3 ((𝜑𝑋𝐴) → (𝐹𝑋) = 𝐶)
2216, 20, 21syl2anc 411 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐹𝑋) = 𝐶)
23 inss2 3371 . . . . 5 (𝐴𝐵) ⊆ 𝐵
245, 23eqsstrri 3203 . . . 4 𝑆𝐵
2524, 19sselid 3168 . . 3 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝑋𝐵)
26 ofrval.7 . . 3 ((𝜑𝑋𝐵) → (𝐺𝑋) = 𝐷)
2716, 25, 26syl2anc 411 . 2 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → (𝐺𝑋) = 𝐷)
2815, 22, 273brtr3d 4049 1 ((𝜑𝐹𝑟 𝑅𝐺𝑋𝑆) → 𝐶𝑅𝐷)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 980   = wceq 1364  wcel 2160  wral 2468  cin 3143   class class class wbr 4018   Fn wfn 5230  cfv 5235  𝑟 cofr 6106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ofr 6108
This theorem is referenced by:  psrbaglesuppg  13967
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