Theorem oprssdmm 6367

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 6365	. . . . . . 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 2298	. . . . . . . . . 10 ⊢ (𝑢 = (𝐹‘⟨𝑥, 𝑦⟩) → (𝑣 ∈ 𝑢 ↔ 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩)))
9	8	exbidv 1874	. . . . . . . . 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 2617	. . . . . . . . . 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 6055	. . . . . . . . . 10 ⊢ (𝑥𝐹𝑦) = (𝐹‘⟨𝑥, 𝑦⟩)
14		oprssdmm.cl	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) ∈ 𝑆)
15	13, 14	eqeltrrid 2322	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝐹‘⟨𝑥, 𝑦⟩) ∈ 𝑆)
16	9, 12, 15	rspcdva 2928	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → ∃𝑣 𝑣 ∈ (𝐹‘⟨𝑥, 𝑦⟩))
17		relelfvdm 5704	. . . . . . . . . 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 1868	. . . . . . . 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 2626	. . . . . 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 3885	. . . . . . 7 ⊢ (𝑥 = (1st ‘𝑧) → ⟨𝑥, 𝑦⟩ = ⟨(1st ‘𝑧), 𝑦⟩)
24	23	eleq1d 2303	. . . . . 6 ⊢ (𝑥 = (1st ‘𝑧) → (⟨𝑥, 𝑦⟩ ∈ dom 𝐹 ↔ ⟨(1st ‘𝑧), 𝑦⟩ ∈ dom 𝐹))
25		opeq2 3886	. . . . . . 7 ⊢ (𝑦 = (2nd ‘𝑧) → ⟨(1st ‘𝑧), 𝑦⟩ = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
26	25	eleq1d 2303	. . . . . 6 ⊢ (𝑦 = (2nd ‘𝑧) → (⟨(1st ‘𝑧), 𝑦⟩ ∈ dom 𝐹 ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ dom 𝐹))
27	24, 26	rspc2va 2937	. . . . 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 2311	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝑆 × 𝑆)) → 𝑧 ∈ dom 𝐹)
30	29	ex 115	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝑆 × 𝑆) → 𝑧 ∈ dom 𝐹))
31	30	ssrdv 3246	1 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522   ⊆ wss 3213  ⟨cop 3694   × cxp 4749  dom cdm 4751  Rel wrel 4756  ‘cfv 5354  (class class class)co 6052  1^st c1st 6334  2^nd c2nd 6335

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fv 5362  df-ov 6055  df-1st 6336  df-2nd 6337

This theorem is referenced by:  axaddf  8188  axmulf  8189