Theorem offval3 8005

Description: General value of (𝐹 ∘_f 𝑅𝐺) with no assumptions on functionality of 𝐹 and 𝐺. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Assertion

Ref	Expression
offval3	⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))

Distinct variable groups:   𝑥,𝐹   𝑥,𝐺   𝑥,𝑉   𝑥,𝑊   𝑥,𝑅

Proof of Theorem offval3

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3498	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ V)
3		elex 3498	. . 3 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
4	3	adantl 481	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V)
5		dmexg 7923	. . . 4 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
6		inex1g 5324	. . . 4 ⊢ (dom 𝐹 ∈ V → (dom 𝐹 ∩ dom 𝐺) ∈ V)
7		mptexg 7240	. . . 4 ⊢ ((dom 𝐹 ∩ dom 𝐺) ∈ V → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V)
8	5, 6, 7	3syl 18	. . 3 ⊢ (𝐹 ∈ 𝑉 → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V)
9	8	adantr 480	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V)
10		dmeq 5916	. . . . 5 ⊢ (𝑎 = 𝐹 → dom 𝑎 = dom 𝐹)
11		dmeq 5916	. . . . 5 ⊢ (𝑏 = 𝐺 → dom 𝑏 = dom 𝐺)
12	10, 11	ineqan12d 4229	. . . 4 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → (dom 𝑎 ∩ dom 𝑏) = (dom 𝐹 ∩ dom 𝐺))
13		fveq1 6905	. . . . 5 ⊢ (𝑎 = 𝐹 → (𝑎‘𝑥) = (𝐹‘𝑥))
14		fveq1 6905	. . . . 5 ⊢ (𝑏 = 𝐺 → (𝑏‘𝑥) = (𝐺‘𝑥))
15	13, 14	oveqan12d 7449	. . . 4 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → ((𝑎‘𝑥)𝑅(𝑏‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
16	12, 15	mpteq12dv 5238	. . 3 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎‘𝑥)𝑅(𝑏‘𝑥))) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
17		df-of 7696	. . 3 ⊢ ∘f 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎‘𝑥)𝑅(𝑏‘𝑥))))
18	16, 17	ovmpoga 7586	. 2 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V ∧ (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))) ∈ V) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))
19	2, 4, 9, 18	syl3anc 1370	1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ (dom 𝐹 ∩ dom 𝐺) ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  Vcvv 3477   ∩ cin 3961   ↦ cmpt 5230  dom cdm 5688  ‘cfv 6562  (class class class)co 7430   ∘_f cof 7694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696

This theorem is referenced by:  offres  8006  offsplitfpar  8142  ofco2  22472  dvsinax  45868  dvcosax  45881  fdivval  48388