Theorem off2 33114

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 6703	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		off2.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
4	3	ffnd 6703	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
5		off2.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		off2.5	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		eqid 2760	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵)
8		eqidd 2761	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧))
9		eqidd 2761	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧))
10	2, 4, 5, 6, 7, 8, 9	offval 7687	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
11		off2.6	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶)
12	11	mpteq1d 5195	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
13	10, 12	eqtrd 2795	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
14	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝑆)
15		inss1 4182	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
16	11, 15	eqsstrrdi 3976	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
17	16	sselda 3931	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴)
18	14, 17	ffvelcdmd 7078	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆)
19	3	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐺:𝐵⟶𝑇)
20		inss2 4183	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
21	11, 20	eqsstrrdi 3976	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
22	21	sselda 3931	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐵)
23	19, 22	ffvelcdmd 7078	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇)
24		off2.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
25	24	ralrimivva 3205	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
26	25	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
27		ovrspc2v 7439	. . 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 851	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)
29	13, 28	fmpt3d 7109	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∩ cin 3898   ↦ cmpt 5186  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413   ∘_f cof 7676

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7678

This theorem is used by: (None)