Theorem off2 32719

Description: The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)

Hypotheses

Ref	Expression
off2.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
off2.2	⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
off2.3	⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
off2.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
off2.5	⊢ (𝜑 → 𝐵 ∈ 𝑊)
off2.6	⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶)

Assertion

Ref	Expression
off2	⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈)

Distinct variable groups:   𝑦,𝐺   𝑥,𝑦,𝜑   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑈,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐺(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem off2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		off2.2	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
2	1	ffnd 6663	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		off2.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
4	3	ffnd 6663	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
5		off2.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		off2.5	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		eqid 2736	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵)
8		eqidd 2737	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧))
9		eqidd 2737	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧))
10	2, 4, 5, 6, 7, 8, 9	offval 7631	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
11		off2.6	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶)
12	11	mpteq1d 5188	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
13	10, 12	eqtrd 2771	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
14	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝑆)
15		inss1 4189	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
16	11, 15	eqsstrrdi 3979	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
17	16	sselda 3933	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴)
18	14, 17	ffvelcdmd 7030	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆)
19	3	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐺:𝐵⟶𝑇)
20		inss2 4190	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
21	11, 20	eqsstrrdi 3979	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
22	21	sselda 3933	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐵)
23	19, 22	ffvelcdmd 7030	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇)
24		off2.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
25	24	ralrimivva 3179	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
26	25	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
27		ovrspc2v 7384	. . 3 ⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ (𝐺‘𝑧) ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)
28	18, 23, 26, 27	syl21anc 837	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)
29	13, 28	fmpt3d 7061	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051   ∩ cin 3900   ↦ cmpt 5179  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ∘_f cof 7620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-of 7622

This theorem is referenced by: (None)