Theorem off2 30402

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 6488	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		off2.3	. . . . 5 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
4	3	ffnd 6488	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐵)
5		off2.4	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		off2.5	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
7		eqid 2798	. . . 4 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵)
8		eqidd 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧))
9		eqidd 2799	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧))
10	2, 4, 5, 6, 7, 8, 9	offval 7396	. . 3 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
11		off2.6	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶)
12	11	mpteq1d 5119	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
13	10, 12	eqtrd 2833	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
14	1	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝑆)
15		inss1 4155	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
16	11, 15	eqsstrrdi 3970	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
17	16	sselda 3915	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴)
18	14, 17	ffvelrnd 6829	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆)
19	3	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐺:𝐵⟶𝑇)
20		inss2 4156	. . . . . 6 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
21	11, 20	eqsstrrdi 3970	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
22	21	sselda 3915	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐵)
23	19, 22	ffvelrnd 6829	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇)
24		off2.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
25	24	ralrimivva 3156	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
26	25	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
27		ovrspc2v 7161	. . 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 836	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)
29	13, 28	fmpt3d 6857	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺):𝐶⟶𝑈)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∩ cin 3880   ↦ cmpt 5110  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∘_f cof 7387

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389

This theorem is referenced by: (None)