Theorem ofrfval 7627

Description: Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-Jul-2014.)

Hypotheses

Ref	Expression
offval.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
offval.2	⊢ (𝜑 → 𝐺 Fn 𝐵)
offval.3	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offval.4	⊢ (𝜑 → 𝐵 ∈ 𝑊)
offval.5	⊢ (𝐴 ∩ 𝐵) = 𝑆
offval.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
offval.7	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)

Assertion

Ref	Expression
ofrfval	⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 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

Step	Hyp	Ref	Expression
1		offval.1	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		offval.2	. 2 ⊢ (𝜑 → 𝐺 Fn 𝐵)
3		offval.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4	1, 3	fnexd 7158	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
5		offval.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6	2, 5	fnexd 7158	. 2 ⊢ (𝜑 → 𝐺 ∈ V)
7		offval.5	. 2 ⊢ (𝐴 ∩ 𝐵) = 𝑆
8		offval.6	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐶)
9		offval.7	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = 𝐷)
10	1, 2, 4, 6, 7, 8, 9	ofrfvalg 7625	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  Vcvv 3438   ∩ cin 3904   class class class wbr 5095   Fn wfn 6481  ‘cfv 6486   ∘_r cofr 7616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ofr 7618

This theorem is referenced by:  ofrval  7629  ofrfval2  7638  caofref  7648  caofrss  7656  caoftrn  7658  ofsubge0  12145  psrbagcon  21850  psrbagleadd1  21853  psrlidm  21887  psdmul  22069  0plef  25589  0pledm  25590  itg1ge0  25603  mbfi1fseqlem5  25636  xrge0f  25648  itg2ge0  25652  itg2lea  25661  itg2splitlem  25665  itg2monolem1  25667  itg2mono  25670  itg2i1fseqle  25671  itg2i1fseq  25672  itg2addlem  25675  itg2cnlem1  25678  itg2addnclem  37650