Theorem basprssdmsets 16132

Description: The pair of the base index and another index is a subset of the domain of the structure obtained by replacing/adding a slot at the other index in a structure having a base slot. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)

Hypotheses

Ref	Expression
basprssdmsets.s	⊢ (𝜑 → 𝑆 Struct 𝑋)
basprssdmsets.i	⊢ (𝜑 → 𝐼 ∈ 𝑈)
basprssdmsets.w	⊢ (𝜑 → 𝐸 ∈ 𝑊)
basprssdmsets.b	⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)

Assertion

Ref	Expression
basprssdmsets	⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Proof of Theorem basprssdmsets

Step	Hyp	Ref	Expression
1		basprssdmsets.b	. . . . 5 ⊢ (𝜑 → (Base‘ndx) ∈ dom 𝑆)
2	1	orcd 862	. . . 4 ⊢ (𝜑 → ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
3		elun 3904	. . . 4 ⊢ ((Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}) ↔ ((Base‘ndx) ∈ dom 𝑆 ∨ (Base‘ndx) ∈ {𝐼}))
4	2, 3	sylibr 224	. . 3 ⊢ (𝜑 → (Base‘ndx) ∈ (dom 𝑆 ∪ {𝐼}))
5		basprssdmsets.i	. . . . . 6 ⊢ (𝜑 → 𝐼 ∈ 𝑈)
6		snidg 4345	. . . . . 6 ⊢ (𝐼 ∈ 𝑈 → 𝐼 ∈ {𝐼})
7	5, 6	syl 17	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ {𝐼})
8	7	olcd 863	. . . 4 ⊢ (𝜑 → (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
9		elun 3904	. . . 4 ⊢ (𝐼 ∈ (dom 𝑆 ∪ {𝐼}) ↔ (𝐼 ∈ dom 𝑆 ∨ 𝐼 ∈ {𝐼}))
10	8, 9	sylibr 224	. . 3 ⊢ (𝜑 → 𝐼 ∈ (dom 𝑆 ∪ {𝐼}))
11	4, 10	prssd 4488	. 2 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ (dom 𝑆 ∪ {𝐼}))
12		basprssdmsets.s	. . . 4 ⊢ (𝜑 → 𝑆 Struct 𝑋)
13		structex 16075	. . . 4 ⊢ (𝑆 Struct 𝑋 → 𝑆 ∈ V)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
15		basprssdmsets.w	. . 3 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
16		setsdm 16099	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝐸 ∈ 𝑊) → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
17	14, 15, 16	syl2anc 573	. 2 ⊢ (𝜑 → dom (𝑆 sSet ⟨𝐼, 𝐸⟩) = (dom 𝑆 ∪ {𝐼}))
18	11, 17	sseqtr4d 3791	1 ⊢ (𝜑 → {(Base‘ndx), 𝐼} ⊆ dom (𝑆 sSet ⟨𝐼, 𝐸⟩))

Syntax hints:   → wi 4   ∨ wo 836   = wceq 1631   ∈ wcel 2145  Vcvv 3351   ∪ cun 3721   ⊆ wss 3723  {csn 4316  {cpr 4318  ⟨cop 4322   class class class wbr 4786  dom cdm 5249  ‘cfv 6031  (class class class)co 6793   Struct cstr 16060  ndxcnx 16061   sSet csts 16062  Basecbs 16064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-struct 16066  df-sets 16071

This theorem is referenced by:  setsvtx  26148  setsiedg  26149