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Theorem ofrfval 7407
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 6975 . . . 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 6975 . . . 4 ((𝐺 Fn 𝐵𝐵𝑊) → 𝐺 ∈ V)
85, 6, 7syl2anc 584 . . 3 (𝜑𝐺 ∈ V)
9 dmeq 5771 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
10 dmeq 5771 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
119, 10ineqan12d 4195 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
12 fveq1 6666 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
13 fveq1 6666 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
1412, 13breqan12d 5079 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
1511, 14raleqbidv 3407 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥) ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
16 df-ofr 7400 . . . 4 r 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥)}
1715, 16brabga 5418 . . 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 6453 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
205fndmd 6453 . . . . 5 (𝜑 → dom 𝐺 = 𝐵)
2119, 20ineq12d 4194 . . . 4 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
22 offval.5 . . . 4 (𝐴𝐵) = 𝑆
2321, 22syl6eq 2877 . . 3 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
2423raleqdv 3421 . 2 (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
25 inss1 4209 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
2622, 25eqsstrri 4006 . . . . . 6 𝑆𝐴
2726sseli 3967 . . . . 5 (𝑥𝑆𝑥𝐴)
28 offval.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
2927, 28sylan2 592 . . . 4 ((𝜑𝑥𝑆) → (𝐹𝑥) = 𝐶)
30 inss2 4210 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
3122, 30eqsstrri 4006 . . . . . 6 𝑆𝐵
3231sseli 3967 . . . . 5 (𝑥𝑆𝑥𝐵)
33 offval.7 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3432, 33sylan2 592 . . . 4 ((𝜑𝑥𝑆) → (𝐺𝑥) = 𝐷)
3529, 34breq12d 5076 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ 𝐶𝑅𝐷))
3635ralbidva 3201 . 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 1530  wcel 2107  wral 3143  Vcvv 3500  cin 3939   class class class wbr 5063  dom cdm 5554   Fn wfn 6347  cfv 6352  r cofr 7398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ofr 7400
This theorem is referenced by:  ofrval  7409  ofrfval2  7417  caofref  7425  caofrss  7432  caoftrn  7434  ofsubge0  11626  pwsle  16755  pwsleval  16756  psrbaglesupp  20067  psrbagcon  20070  psrbaglefi  20071  psrlidm  20102  0plef  24188  0pledm  24189  itg1ge0  24202  mbfi1fseqlem5  24235  xrge0f  24247  itg2ge0  24251  itg2lea  24260  itg2splitlem  24264  itg2monolem1  24266  itg2mono  24269  itg2i1fseqle  24270  itg2i1fseq  24271  itg2addlem  24274  itg2cnlem1  24277  itg2addnclem  34810
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