Theorem oprssdmm 6139

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 6137	. . . . . . 7 ⊢ (𝑧 ∈ (𝑆 × 𝑆) ↔ (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑆)))
2	1	biimpi 119	. . . . . 6 ⊢ (𝑧 ∈ (𝑆 × 𝑆) → (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑆)))
3	2	adantl 275	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → (𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∧ ((1st ‘𝑧) ∈ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑆)))
4	3	simpld 111	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
5	3	simprd 113	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → ((1st ‘𝑧) ∈ 𝑆 ∧ (2nd ‘𝑧) ∈ 𝑆))
6		oprssdmm.f	. . . . . . . . 9 ⊢ (𝜑 → Rel 𝐹)
7	6	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → Rel 𝐹)
8		eleq2 2230	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘⟨𝑥, 𝑦⟩) → (𝑣 ∈ 𝑢 ↔ 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)))
9	8	exbidv 1813	. . . . . . . . 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 2539	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑢 ∈ 𝑆 ∃𝑣 𝑣 ∈ 𝑢)
12	11	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∀𝑢 ∈ 𝑆 ∃𝑣 𝑣 ∈ 𝑢)
13		df-ov 5845	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
14		oprssdmm.cl	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
15	13, 14	eqeltrrid 2254	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
16	9, 12, 15	rspcdva 2835	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∃𝑣 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩))
17		relelfvdm 5518	. . . . . . . . . 10 ⊢ ((Rel 𝐹 ∧ 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
18	17	ex 114	. . . . . . . . 9 ⊢ (Rel 𝐹 → (𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ ∈ dom 𝐹))
19	18	exlimdv 1807	. . . . . . . 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 2548	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
22	21	adantr 274	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 ⟨𝑥, 𝑦⟩ ∈ dom 𝐹)
23		opeq1 3758	. . . . . . 7 ⊢ (𝑥 = (1st ‘𝑧) → ⟨𝑥, 𝑦⟩ = ⟨(1st ‘𝑧), 𝑦⟩)
24	23	eleq1d 2235	. . . . . 6 ⊢ (𝑥 = (1st ‘𝑧) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨(1st ‘𝑧), 𝑦⟩ ∈ dom 𝐹))
25		opeq2 3759	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨(1st ‘𝑧), 𝑦⟩ = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
26	25	eleq1d 2235	. . . . . 6 ⊢ (𝑦 = (2nd ‘𝑧) → (⟨(1st ‘𝑧), 𝑦⟩ ∈ dom 𝐹 ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ dom 𝐹))
27	24, 26	rspc2va 2844	. . . . 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 409	. . . 4 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ dom 𝐹)
29	4, 28	eqeltrd 2243	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → 𝑧 ∈ dom 𝐹)
30	29	ex 114	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝑆 × 𝑆) → 𝑧 ∈ dom 𝐹))
31	30	ssrdv 3148	1 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444   ⊆ wss 3116  ⟨cop 3579   × cxp 4602  dom cdm 4604  Rel wrel 4609  ‘cfv 5188  (class class class)co 5842  1^st c1st 6106  2^nd c2nd 6107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-iota 5153  df-fun 5190  df-fv 5196  df-ov 5845  df-1st 6108  df-2nd 6109

This theorem is referenced by:  axaddf  7809  axmulf  7810