funvtxdm2vald - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > funvtxdm2vald		GIF version

Theorem funvtxdm2vald 16477

Description: The set of vertices of an extensible structure with (at least) two slots. (Contributed by AV, 22-Sep-2020.) (Revised by Jim Kingdon, 11-Dec-2025.)

**Hypotheses**
Ref	Expression
funvtxdm2val.a	⊢ 𝐴 ∈ V
funvtxdm2val.b	⊢ 𝐵 ∈ V
funvtxdm2vald.g	⊢ (𝜑 → 𝐺 ∈ 𝑋)
funvtxdm2vald.fun	⊢ (𝜑 → Fun (𝐺 ∖ {∅}))
funvtxdm2vald.ne	⊢ (𝜑 → 𝐴 ≠ 𝐵)
funvtxdm2vald.dm	⊢ (𝜑 → {𝐴, 𝐵} ⊆ dom 𝐺)

**Assertion**
Ref	Expression
funvtxdm2vald	⊢ (𝜑 → (Vtx‘𝐺) = (Base‘𝐺))

**Proof of Theorem funvtxdm2vald**
Step	Hyp	Ref	Expression
1		funvtxdm2vald.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑋)
2		vtxvalg 16462	. . 3 ⊢ (𝐺 ∈ 𝑋 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → (Vtx‘𝐺) = if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)))
4		funvtxdm2vald.fun	. . . 4 ⊢ (𝜑 → Fun (𝐺 ∖ {∅}))
5		funvtxdm2vald.ne	. . . 4 ⊢ (𝜑 → 𝐴 ≠ 𝐵)
6		funvtxdm2vald.dm	. . . 4 ⊢ (𝜑 → {𝐴, 𝐵} ⊆ dom 𝐺)
7		funvtxdm2val.a	. . . . 5 ⊢ 𝐴 ∈ V
8		funvtxdm2val.b	. . . . 5 ⊢ 𝐵 ∈ V
9	7, 8	fun2dmnop0 11324	. . . 4 ⊢ ((Fun (𝐺 ∖ {∅}) ∧ 𝐴 ≠ 𝐵 ∧ {𝐴, 𝐵} ⊆ dom 𝐺) → ¬ 𝐺 ∈ (V × V))
10	4, 5, 6, 9	syl3anc 1278	. . 3 ⊢ (𝜑 → ¬ 𝐺 ∈ (V × V))
11	10	iffalsed 3650	. 2 ⊢ (𝜑 → if(𝐺 ∈ (V × V), (1^st ‘𝐺), (Base‘𝐺)) = (Base‘𝐺))
12	3, 11	eqtrd 2271	1 ⊢ (𝜑 → (Vtx‘𝐺) = (Base‘𝐺))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 Vcvv 2821 ∖ cdif 3217 ⊆ wss 3220 ∅c0 3520 ifcif 3638 {csn 3709 {cpr 3710 × cxp 4774 dom cdm 4776 Fun wfun 5374 ‘cfv 5380 1^st c1st 6376 Basecbs 13410 Vtxcvtx 16458

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-cnex 8276 ax-resscn 8277 ax-1re 8279 ax-addrcl 8282

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-iord 4513 df-on 4515 df-suc 4518 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-1st 6378 df-1o 6691 df-2o 6692 df-en 7027 df-dom 7028 df-inn 9314 df-ndx 13413 df-slot 13414 df-base 13416 df-vtx 16460

This theorem is used by: funvtxval0d 16479 funvtxvalg 16482

W3C validator