Theorem ofrfval 7666

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 7198	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
5		offval.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6	2, 5	fnexd 7198	. 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 7664	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ∩ cin 3903   class class class wbr 5099   Fn wfn 6512  ‘cfv 6517   ∘_r cofr 7655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ofr 7657

This theorem is referenced by:  ofrval  7668  ofrfval2  7677  caofref  7687  caofrss  7695  caoftrn  7697  ofsubge0  12191  psrbagcon  21957  psrbagleadd1  21960  psrlidm  21993  psdmul  22211  0plef  25714  0pledm  25715  itg1ge0  25728  mbfi1fseqlem5  25761  xrge0f  25773  itg2ge0  25777  itg2lea  25786  itg2splitlem  25790  itg2monolem1  25792  itg2mono  25795  itg2i1fseqle  25796  itg2i1fseq  25797  itg2addlem  25800  itg2cnlem1  25803  fnfvor  32761  ofrco  32762  selvply1rhmlemb  33777  esplyind  33833  itg2addnclem  38134