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Theorem ofrfval 6947
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 6522 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
41, 2, 3syl2anc 694 . . 3 (𝜑𝐹 ∈ V)
5 offval.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 offval.4 . . . 4 (𝜑𝐵𝑊)
7 fnex 6522 . . . 4 ((𝐺 Fn 𝐵𝐵𝑊) → 𝐺 ∈ V)
85, 6, 7syl2anc 694 . . 3 (𝜑𝐺 ∈ V)
9 dmeq 5356 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
10 dmeq 5356 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
119, 10ineqan12d 3849 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
12 fveq1 6228 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
13 fveq1 6228 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
1412, 13breqan12d 4701 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
1511, 14raleqbidv 3182 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥) ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
16 df-ofr 6940 . . . 4 𝑟 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥)}
1715, 16brabga 5018 . . 3 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
184, 8, 17syl2anc 694 . 2 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
19 fndm 6028 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
201, 19syl 17 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
21 fndm 6028 . . . . . 6 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
225, 21syl 17 . . . . 5 (𝜑 → dom 𝐺 = 𝐵)
2320, 22ineq12d 3848 . . . 4 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
24 offval.5 . . . 4 (𝐴𝐵) = 𝑆
2523, 24syl6eq 2701 . . 3 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
2625raleqdv 3174 . 2 (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
27 inss1 3866 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
2824, 27eqsstr3i 3669 . . . . . 6 𝑆𝐴
2928sseli 3632 . . . . 5 (𝑥𝑆𝑥𝐴)
30 offval.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
3129, 30sylan2 490 . . . 4 ((𝜑𝑥𝑆) → (𝐹𝑥) = 𝐶)
32 inss2 3867 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
3324, 32eqsstr3i 3669 . . . . . 6 𝑆𝐵
3433sseli 3632 . . . . 5 (𝑥𝑆𝑥𝐵)
35 offval.7 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3634, 35sylan2 490 . . . 4 ((𝜑𝑥𝑆) → (𝐺𝑥) = 𝐷)
3731, 36breq12d 4698 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ 𝐶𝑅𝐷))
3837ralbidva 3014 . 2 (𝜑 → (∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
3918, 26, 383bitrd 294 1 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cin 3606   class class class wbr 4685  dom cdm 5143   Fn wfn 5921  cfv 5926  𝑟 cofr 6938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ofr 6940
This theorem is referenced by:  ofrval  6949  ofrfval2  6957  caofref  6965  caofrss  6972  caoftrn  6974  ofsubge0  11057  pwsle  16199  pwsleval  16200  psrbaglesupp  19416  psrbagcon  19419  psrbaglefi  19420  psrlidm  19451  0plef  23484  0pledm  23485  itg1ge0  23498  mbfi1fseqlem5  23531  xrge0f  23543  itg2ge0  23547  itg2lea  23556  itg2splitlem  23560  itg2monolem1  23562  itg2mono  23565  itg2i1fseqle  23566  itg2i1fseq  23567  itg2addlem  23570  itg2cnlem1  23573  itg2addnclem  33591
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