Theorem oprssdmm 6334

Description: Domain of closure of an operation. (Contributed by Jim Kingdon, 23-Oct-2023.)

Hypotheses

Ref	Expression
oprssdmm.m	⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆) → ∃𝑣 𝑣 ∈ 𝑢)
oprssdmm.cl	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
oprssdmm.f	⊢ (𝜑 → Rel 𝐹)

Assertion

Ref	Expression
oprssdmm	⊢ (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)

Distinct variable groups:   𝑢,𝐹,𝑣,𝑥,𝑦   𝑢,𝑆,𝑥,𝑦   𝜑,𝑢,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑣)   𝑆(𝑣)

Proof of Theorem oprssdmm

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp6 6332	. . . . . . 7 ⊢ (𝑧 ∈ (𝑆 × 𝑆) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑆)))
2	1	biimpi 120	. . . . . 6 ⊢ (𝑧 ∈ (𝑆 × 𝑆) → (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑆)))
3	2	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑆)))
4	3	simpld 112	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
5	3	simprd 114	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → ((1st ‘𝑧) ∈ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑆))
6		oprssdmm.f	. . . . . . . . 9 ⊢ (𝜑 → Rel 𝐹)
7	6	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → Rel 𝐹)
8		eleq2 2295	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘⟨𝑥, 𝑦⟩) → (𝑣 ∈ 𝑢 ↔ 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)))
9	8	exbidv 1873	. . . . . . . . 9 ⊢ (𝑢 = (𝐹‘⟨𝑥, 𝑦⟩) → (∃𝑣 𝑣 ∈ 𝑢 ↔ ∃𝑣 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)))
10		oprssdmm.m	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑢 ∈ 𝑆) → ∃𝑣 𝑣 ∈ 𝑢)
11	10	ralrimiva 2605	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑢 ∈ 𝑆 ∃𝑣 𝑣 ∈ 𝑢)
12	11	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑢 ∈ 𝑆 ∃𝑣 𝑣 ∈ 𝑢)
13		df-ov 6021	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
14		oprssdmm.cl	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
15	13, 14	eqeltrrid 2319	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
16	9, 12, 15	rspcdva 2915	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∃𝑣 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩))
17		relelfvdm 5671	. . . . . . . . . 10 ⊢ ((Rel 𝐹 ∧ 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
18	17	ex 115	. . . . . . . . 9 ⊢ (Rel 𝐹 → (𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
19	18	exlimdv 1867	. . . . . . . 8 ⊢ (Rel 𝐹 → (∃𝑣 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
20	7, 16, 19	sylc 62	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
21	20	ralrimivva 2614	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
22	21	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
23		opeq1 3862	. . . . . . 7 ⊢ (𝑥 = (1st ‘𝑧) → ⟨𝑥, 𝑦⟩ = ⟨(1st ‘𝑧), 𝑦⟩)
24	23	eleq1d 2300	. . . . . 6 ⊢ (𝑥 = (1st ‘𝑧) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨(1st ‘𝑧), 𝑦⟩ ∈ dom 𝐹))
25		opeq2 3863	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨(1st ‘𝑧), 𝑦⟩ = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
26	25	eleq1d 2300	. . . . . 6 ⊢ (𝑦 = (2nd ‘𝑧) → (⟨(1st ‘𝑧), 𝑦⟩ ∈ dom 𝐹 ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ dom 𝐹))
27	24, 26	rspc2va 2924	. . . . 5 ⊢ ((((1st ‘𝑧) ∈ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑆) ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ⟨𝑥, 𝑦⟩ ∈ dom 𝐹) → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ dom 𝐹)
28	5, 22, 27	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ dom 𝐹)
29	4, 28	eqeltrd 2308	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → 𝑧 ∈ dom 𝐹)
30	29	ex 115	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝑆 × 𝑆) → 𝑧 ∈ dom 𝐹))
31	30	ssrdv 3233	1 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510   ⊆ wss 3200  ⟨cop 3672   × cxp 4723  dom cdm 4725  Rel wrel 4730  ‘cfv 5326  (class class class)co 6018  1^st c1st 6301  2^nd c2nd 6302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6021  df-1st 6303  df-2nd 6304

This theorem is referenced by:  axaddf  8088  axmulf  8089