Theorem ofrfval 7703

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 7224	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
5		offval.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6	2, 5	fnexd 7224	. 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 7701	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ∩ cin 4089   class class class wbr 5102   Fn wfn 6533  ‘cfv 6538   ∘_r cofr 7692

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ofr 7694

This theorem is used by:  ofrval  7705  ofrfval2  7714  caofref  7724  caofrss  7732  caoftrn  7734  ofsubge0  12322  psrbagcon  22238  psrbagleadd1  22241  psrlidm  22274  psdmul  22492  0plef  25998  0pledm  25999  itg1ge0  26012  mbfi1fseqlem5  26045  xrge0f  26057  itg2ge0  26061  itg2lea  26070  itg2splitlem  26074  itg2monolem1  26076  itg2mono  26079  itg2i1fseqle  26080  itg2i1fseq  26081  itg2addlem  26084  itg2cnlem1  26087  fnfvor  33250  ofrco  33251  selvply1rhmlemb  34197  esplyind  34253  itg2addnclem  38682