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Theorem ofrfval 7406
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 6971 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
41, 2, 3syl2anc 584 . . 3 (𝜑𝐹 ∈ V)
5 offval.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 offval.4 . . . 4 (𝜑𝐵𝑊)
7 fnex 6971 . . . 4 ((𝐺 Fn 𝐵𝐵𝑊) → 𝐺 ∈ V)
85, 6, 7syl2anc 584 . . 3 (𝜑𝐺 ∈ V)
9 dmeq 5765 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
10 dmeq 5765 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
119, 10ineqan12d 4188 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
12 fveq1 6662 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
13 fveq1 6662 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
1412, 13breqan12d 5073 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
1511, 14raleqbidv 3399 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥) ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
16 df-ofr 7399 . . . 4 r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥)}
1715, 16brabga 5412 . . 3 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
184, 8, 17syl2anc 584 . 2 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
191fndmd 6449 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
205fndmd 6449 . . . . 5 (𝜑 → dom 𝐺 = 𝐵)
2119, 20ineq12d 4187 . . . 4 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
22 offval.5 . . . 4 (𝐴𝐵) = 𝑆
2321, 22syl6eq 2869 . . 3 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
2423raleqdv 3413 . 2 (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
25 inss1 4202 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
2622, 25eqsstrri 3999 . . . . . 6 𝑆𝐴
2726sseli 3960 . . . . 5 (𝑥𝑆𝑥𝐴)
28 offval.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
2927, 28sylan2 592 . . . 4 ((𝜑𝑥𝑆) → (𝐹𝑥) = 𝐶)
30 inss2 4203 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
3122, 30eqsstrri 3999 . . . . . 6 𝑆𝐵
3231sseli 3960 . . . . 5 (𝑥𝑆𝑥𝐵)
33 offval.7 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3432, 33sylan2 592 . . . 4 ((𝜑𝑥𝑆) → (𝐺𝑥) = 𝐷)
3529, 34breq12d 5070 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ 𝐶𝑅𝐷))
3635ralbidva 3193 . 2 (𝜑 → (∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
3718, 24, 363bitrd 306 1 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cin 3932   class class class wbr 5057  dom cdm 5548   Fn wfn 6343  cfv 6348  r cofr 7397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ofr 7399
This theorem is referenced by:  ofrval  7408  ofrfval2  7416  caofref  7424  caofrss  7431  caoftrn  7433  ofsubge0  11625  pwsle  16753  pwsleval  16754  psrbaglesupp  20076  psrbagcon  20079  psrbaglefi  20080  psrlidm  20111  0plef  24200  0pledm  24201  itg1ge0  24214  mbfi1fseqlem5  24247  xrge0f  24259  itg2ge0  24263  itg2lea  24272  itg2splitlem  24276  itg2monolem1  24278  itg2mono  24281  itg2i1fseqle  24282  itg2i1fseq  24283  itg2addlem  24286  itg2cnlem1  24289  itg2addnclem  34824
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