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Theorem ofrfval 6865
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 6441 . . . 4 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
41, 2, 3syl2anc 692 . . 3 (𝜑𝐹 ∈ V)
5 offval.2 . . . 4 (𝜑𝐺 Fn 𝐵)
6 offval.4 . . . 4 (𝜑𝐵𝑊)
7 fnex 6441 . . . 4 ((𝐺 Fn 𝐵𝐵𝑊) → 𝐺 ∈ V)
85, 6, 7syl2anc 692 . . 3 (𝜑𝐺 ∈ V)
9 dmeq 5289 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
10 dmeq 5289 . . . . . 6 (𝑔 = 𝐺 → dom 𝑔 = dom 𝐺)
119, 10ineqan12d 3799 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → (dom 𝑓 ∩ dom 𝑔) = (dom 𝐹 ∩ dom 𝐺))
12 fveq1 6152 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
13 fveq1 6152 . . . . . 6 (𝑔 = 𝐺 → (𝑔𝑥) = (𝐺𝑥))
1412, 13breqan12d 4634 . . . . 5 ((𝑓 = 𝐹𝑔 = 𝐺) → ((𝑓𝑥)𝑅(𝑔𝑥) ↔ (𝐹𝑥)𝑅(𝐺𝑥)))
1511, 14raleqbidv 3144 . . . 4 ((𝑓 = 𝐹𝑔 = 𝐺) → (∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥) ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
16 df-ofr 6858 . . . 4 𝑟 𝑅 = {⟨𝑓, 𝑔⟩ ∣ ∀𝑥 ∈ (dom 𝑓 ∩ dom 𝑔)(𝑓𝑥)𝑅(𝑔𝑥)}
1715, 16brabga 4954 . . 3 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
184, 8, 17syl2anc 692 . 2 (𝜑 → (𝐹𝑟 𝑅𝐺 ↔ ∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥)))
19 fndm 5953 . . . . . 6 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
201, 19syl 17 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
21 fndm 5953 . . . . . 6 (𝐺 Fn 𝐵 → dom 𝐺 = 𝐵)
225, 21syl 17 . . . . 5 (𝜑 → dom 𝐺 = 𝐵)
2320, 22ineq12d 3798 . . . 4 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = (𝐴𝐵))
24 offval.5 . . . 4 (𝐴𝐵) = 𝑆
2523, 24syl6eq 2671 . . 3 (𝜑 → (dom 𝐹 ∩ dom 𝐺) = 𝑆)
2625raleqdv 3136 . 2 (𝜑 → (∀𝑥 ∈ (dom 𝐹 ∩ dom 𝐺)(𝐹𝑥)𝑅(𝐺𝑥) ↔ ∀𝑥𝑆 (𝐹𝑥)𝑅(𝐺𝑥)))
27 inss1 3816 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
2824, 27eqsstr3i 3620 . . . . . 6 𝑆𝐴
2928sseli 3583 . . . . 5 (𝑥𝑆𝑥𝐴)
30 offval.6 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
3129, 30sylan2 491 . . . 4 ((𝜑𝑥𝑆) → (𝐹𝑥) = 𝐶)
32 inss2 3817 . . . . . . 7 (𝐴𝐵) ⊆ 𝐵
3324, 32eqsstr3i 3620 . . . . . 6 𝑆𝐵
3433sseli 3583 . . . . 5 (𝑥𝑆𝑥𝐵)
35 offval.7 . . . . 5 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
3634, 35sylan2 491 . . . 4 ((𝜑𝑥𝑆) → (𝐺𝑥) = 𝐷)
3731, 36breq12d 4631 . . 3 ((𝜑𝑥𝑆) → ((𝐹𝑥)𝑅(𝐺𝑥) ↔ 𝐶𝑅𝐷))
3837ralbidva 2980 . 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 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  cin 3558   class class class wbr 4618  dom cdm 5079   Fn wfn 5847  cfv 5852  𝑟 cofr 6856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ofr 6858
This theorem is referenced by:  ofrval  6867  ofrfval2  6875  caofref  6883  caofrss  6890  caoftrn  6892  ofsubge0  10970  pwsle  16080  pwsleval  16081  psrbaglesupp  19296  psrbagcon  19299  psrbaglefi  19300  psrlidm  19331  0plef  23358  0pledm  23359  itg1ge0  23372  mbfi1fseqlem5  23405  xrge0f  23417  itg2ge0  23421  itg2lea  23430  itg2splitlem  23434  itg2monolem1  23436  itg2mono  23439  itg2i1fseqle  23440  itg2i1fseq  23441  itg2addlem  23444  itg2cnlem1  23447  itg2addnclem  33120
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