Theorem off2 29443

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	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶𝑆)
2	1	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝑆)
3		off2.6	. . . . . . 7 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = 𝐶)
4		inss1 3833	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
5	3, 4	syl6eqssr 3656	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
6	5	sselda 3603	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐴)
7	2, 6	ffvelrnd 6360	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐹‘𝑧) ∈ 𝑆)
8		off2.3	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐵⟶𝑇)
9	8	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐺:𝐵⟶𝑇)
10		inss2 3834	. . . . . . 7 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
11	3, 10	syl6eqssr 3656	. . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
12	11	sselda 3603	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐵)
13	9, 12	ffvelrnd 6360	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐺‘𝑧) ∈ 𝑇)
14		off2.1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)
15	14	ralrimivva 2971	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
16	15	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈)
17		ovrspc2v 6672	. . . 4 ⊢ ((((𝐹‘𝑧) ∈ 𝑆 ∧ (𝐺‘𝑧) ∈ 𝑇) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑇 (𝑥𝑅𝑦) ∈ 𝑈) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)
18	7, 13, 16, 17	syl21anc 1325	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐹‘𝑧)𝑅(𝐺‘𝑧)) ∈ 𝑈)
19		eqid 2622	. . 3 ⊢ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧)))
20	18, 19	fmptd 6385	. 2 ⊢ (𝜑 → (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈)
21		ffn 6045	. . . . . 6 ⊢ (𝐹:𝐴⟶𝑆 → 𝐹 Fn 𝐴)
22	1, 21	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
23		ffn 6045	. . . . . 6 ⊢ (𝐺:𝐵⟶𝑇 → 𝐺 Fn 𝐵)
24	8, 23	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 Fn 𝐵)
25		off2.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
26		off2.5	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
27		eqid 2622	. . . . 5 ⊢ (𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐵)
28		eqidd 2623	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐹‘𝑧))
29		eqidd 2623	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐵) → (𝐺‘𝑧) = (𝐺‘𝑧))
30	22, 24, 25, 26, 27, 28, 29	offval 6904	. . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
31	3	mpteq1d 4738	. . . 4 ⊢ (𝜑 → (𝑧 ∈ (𝐴 ∩ 𝐵) ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
32	30, 31	eqtrd 2656	. . 3 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))))
33	32	feq1d 6030	. 2 ⊢ (𝜑 → ((𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈 ↔ (𝑧 ∈ 𝐶 ↦ ((𝐹‘𝑧)𝑅(𝐺‘𝑧))):𝐶⟶𝑈))
34	20, 33	mpbird 247	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺):𝐶⟶𝑈)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∩ cin 3573   ↦ cmpt 4729   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ∘_𝑓 cof 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by: (None)