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Theorem ofrfval 7632
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
StepHypRef Expression
1 offval.1 . 2 (𝜑𝐹 Fn 𝐴)
2 offval.2 . 2 (𝜑𝐺 Fn 𝐵)
3 offval.3 . . 3 (𝜑𝐴𝑉)
41, 3fnexd 7173 . 2 (𝜑𝐹 ∈ V)
5 offval.4 . . 3 (𝜑𝐵𝑊)
62, 5fnexd 7173 . 2 (𝜑𝐺 ∈ V)
7 offval.5 . 2 (𝐴𝐵) = 𝑆
8 offval.6 . 2 ((𝜑𝑥𝐴) → (𝐹𝑥) = 𝐶)
9 offval.7 . 2 ((𝜑𝑥𝐵) → (𝐺𝑥) = 𝐷)
101, 2, 4, 6, 7, 8, 9ofrfvalg 7630 1 (𝜑 → (𝐹r 𝑅𝐺 ↔ ∀𝑥𝑆 𝐶𝑅𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3065  Vcvv 3448  cin 3914   class class class wbr 5110   Fn wfn 6496  cfv 6501  r cofr 7621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ofr 7623
This theorem is referenced by:  ofrval  7634  ofrfval2  7643  caofref  7651  caofrss  7658  caoftrn  7660  ofsubge0  12159  psrbaglesuppOLD  21343  psrbagcon  21348  psrbagconOLD  21349  psrbaglefiOLD  21351  psrlidm  21388  0plef  25052  0pledm  25053  itg1ge0  25066  mbfi1fseqlem5  25100  xrge0f  25112  itg2ge0  25116  itg2lea  25125  itg2splitlem  25129  itg2monolem1  25131  itg2mono  25134  itg2i1fseqle  25135  itg2i1fseq  25136  itg2addlem  25139  itg2cnlem1  25142  itg2addnclem  36158
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