Theorem ofrfval 7682

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 7221	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
5		offval.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6	2, 5	fnexd 7221	. 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 7680	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059  Vcvv 3472   ∩ cin 3946   class class class wbr 5147   Fn wfn 6537  ‘cfv 6542   ∘_r cofr 7671

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ofr 7673

This theorem is referenced by:  ofrval  7684  ofrfval2  7693  caofref  7701  caofrss  7708  caoftrn  7710  ofsubge0  12215  psrbaglesuppOLD  21697  psrbagcon  21702  psrbagconOLD  21703  psrbaglefiOLD  21705  psrlidm  21742  0plef  25421  0pledm  25422  itg1ge0  25435  mbfi1fseqlem5  25469  xrge0f  25481  itg2ge0  25485  itg2lea  25494  itg2splitlem  25498  itg2monolem1  25500  itg2mono  25503  itg2i1fseqle  25504  itg2i1fseq  25505  itg2addlem  25508  itg2cnlem1  25511  itg2addnclem  36842