Theorem off2 32733

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 6656	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		off2.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
4	3	ffnd 6656	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
5		off2.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		off2.5	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		eqid 2739	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵)
8		eqidd 2740	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧))
9		eqidd 2740	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧))
10	2, 4, 5, 6, 7, 8, 9	offval 7629	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
11		off2.6	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶)
12	11	mpteq1d 5162	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
13	10, 12	eqtrd 2774	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
14	1	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝑆)
15		inss1 4165	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
16	11, 15	eqsstrrdi 3960	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
17	16	sselda 3915	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴)
18	14, 17	ffvelcdmd 7026	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆)
19	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐺:𝐵⟶𝑇)
20		inss2 4166	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
21	11, 20	eqsstrrdi 3960	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
22	21	sselda 3915	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐵)
23	19, 22	ffvelcdmd 7026	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇)
24		off2.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
25	24	ralrimivva 3182	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
26	25	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
27		ovrspc2v 7382	. . 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 843	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)
29	13, 28	fmpt3d 7057	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ∩ cin 3882   ↦ cmpt 5153  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ∘_f cof 7618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620

This theorem is referenced by: (None)