Theorem offval3 7936

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 3463	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐹 ∈ V)
3		elex 3463	. . 3 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
4	3	adantl 481	. 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊) → 𝐺 ∈ V)
5		dmexg 7853	. . . 4 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
6		inex1g 5266	. . . 4 ⊢ (dom 𝐹 ∈ V → (dom 𝐹 ∩ dom 𝐺) ∈ V)
7		mptexg 7177	. . . 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 5860	. . . . 5 ⊢ (𝑎 = 𝐹 → dom 𝑎 = dom 𝐹)
11		dmeq 5860	. . . . 5 ⊢ (𝑏 = 𝐺 → dom 𝑏 = dom 𝐺)
12	10, 11	ineqan12d 4176	. . . 4 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → (dom 𝑎 ∩ dom 𝑏) = (dom 𝐹 ∩ dom 𝐺))
13		fveq1 6841	. . . . 5 ⊢ (𝑎 = 𝐹 → (𝑎‘𝑥) = (𝐹‘𝑥))
14		fveq1 6841	. . . . 5 ⊢ (𝑏 = 𝐺 → (𝑏‘𝑥) = (𝐺‘𝑥))
15	13, 14	oveqan12d 7387	. . . 4 ⊢ ((𝑎 = 𝐹 ∧ 𝑏 = 𝐺) → ((𝑎‘𝑥)𝑅(𝑏‘𝑥)) = ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))
16	12, 15	mpteq12dv 5187	. . 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 7632	. . 3 ⊢ ∘f 𝑅 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑥 ∈ (dom 𝑎 ∩ dom 𝑏) ↦ ((𝑎‘𝑥)𝑅(𝑏‘𝑥))))
18	16, 17	ovmpoga 7522	. 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 1374	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 1542   ∈ wcel 2114  Vcvv 3442   ∩ cin 3902   ↦ cmpt 5181  dom cdm 5632  ‘cfv 6500  (class class class)co 7368   ∘_f cof 7630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-of 7632

This theorem is referenced by:  offres  7937  offsplitfpar  8071  ofco2  22407  dvsinax  46271  dvcosax  46284  fdivval  48899