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Theorem ofrfval 6091
Description: Value of a relation applied to two functions. (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
offval.6 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
offval.7 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
Assertion
Ref Expression
ofrfval (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝑆   𝑥,𝑅
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ofrfval
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 offval.1 . . . 4 (𝜑𝐹 Fn 𝐴)
2 offval.3 . . . 4 (𝜑𝐴𝑉)
3 fnex 5739 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
41, 2, 3syl2anc 411 . . 3 (𝜑𝐹 ∈ V)
5 offval.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 offval.4 . . . 4 (𝜑𝐵𝑊)
7 fnex 5739 . . . 4 ((𝐺 Fn 𝐵𝐵𝑊) → 𝐺 ∈ V)
85, 6, 7syl2anc 411 . . 3 (𝜑𝐺 ∈ V)
9 dmeq 4828 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
10 dmeq 4828 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
119, 10ineqan12d 3339 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
12 fveq1 5515 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
13 fveq1 5515 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
1412, 13breqan12d 4020 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
1511, 14raleqbidv 2685 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥) ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
16 df-ofr 6084 . . . 4 𝑟 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥)}
1715, 16brabga 4265 . . 3 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
184, 8, 17syl2anc 411 . 2 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
19 fndm 5316 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
201, 19syl 14 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
21 fndm 5316 . . . . . 6 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
225, 21syl 14 . . . . 5 (𝜑 → dom 𝐺 = 𝐵)
2320, 22ineq12d 3338 . . . 4 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
24 offval.5 . . . 4 (𝐴𝐵) = 𝑆
2523, 24eqtrdi 2226 . . 3 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
2625raleqdv 2679 . 2 (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
27 inss1 3356 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
2824, 27eqsstrri 3189 . . . . . 6 𝑆𝐴
2928sseli 3152 . . . . 5 (𝑥𝑆𝑥𝐴)
30 offval.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
3129, 30sylan2 286 . . . 4 ((𝜑𝑥𝑆) → (𝐹𝑥) = 𝐶)
32 inss2 3357 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
3324, 32eqsstrri 3189 . . . . . 6 𝑆𝐵
3433sseli 3152 . . . . 5 (𝑥𝑆𝑥𝐵)
35 offval.7 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3634, 35sylan2 286 . . . 4 ((𝜑𝑥𝑆) → (𝐺𝑥) = 𝐷)
3731, 36breq12d 4017 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ 𝐶𝑅𝐷))
3837ralbidva 2473 . 2 (𝜑 → (∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
3918, 26, 383bitrd 214 1 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  Vcvv 2738  cin 3129   class class class wbr 4004  dom cdm 4627   Fn wfn 5212  cfv 5217  𝑟 cofr 6082
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ofr 6084
This theorem is referenced by:  ofrval  6093  ofrfval2  6099  caofref  6104  caofrss  6107  caoftrn  6108
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