Theorem ofrfval 7730

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 7250	. 2 ⊢ (𝜑 → 𝐹 ∈ V)
5		offval.4	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
6	2, 5	fnexd 7250	. 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 7728	1 ⊢ (𝜑 → (𝐹 ∘r 𝑅𝐺 ↔ ∀𝑥 ∈ 𝑆 𝐶𝑅𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ∩ cin 4092   class class class wbr 5106   Fn wfn 6556  ‘cfv 6561   ∘_r cofr 7719

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ofr 7721

This theorem is used by:  ofrval  7732  ofrfval2  7741  caofref  7751  caofrss  7759  caoftrn  7761  ofsubge0  12349  psrbagcon  22266  psrbagleadd1  22269  psrlidm  22302  psdmul  22520  0plef  26026  0pledm  26027  itg1ge0  26040  mbfi1fseqlem5  26073  xrge0f  26085  itg2ge0  26089  itg2lea  26098  itg2splitlem  26102  itg2monolem1  26104  itg2mono  26107  itg2i1fseqle  26108  itg2i1fseq  26109  itg2addlem  26112  itg2cnlem1  26115  fnfvor  33278  ofrco  33279  selvply1rhmlemb  34234  esplyind  34290  itg2addnclem  38794