ennnfonelem0 - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > ennnfonelem0		GIF version

Theorem ennnfonelem0 12992

Description: Lemma for ennnfone 13012. Initial value. (Contributed by Jim Kingdon, 15-Jul-2023.)

**Hypotheses**
Ref	Expression
ennnfonelemh.dceq	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
ennnfonelemh.f	⊢ (𝜑 → 𝐹:ω–onto→𝐴)
ennnfonelemh.ne	⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
ennnfonelemh.g	⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑_pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
ennnfonelemh.n	⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
ennnfonelemh.j	⊢ 𝐽 = (𝑥 ∈ ℕ₀ ↦ if(𝑥 = 0, ∅, (^◡𝑁‘(𝑥 − 1))))
ennnfonelemh.h	⊢ 𝐻 = seq0(𝐺, 𝐽)

**Assertion**
Ref	Expression
ennnfonelem0	⊢ (𝜑 → (𝐻‘0) = ∅)

Distinct variable groups: 𝐴,𝑗,𝑥,𝑦 𝑥,𝐹,𝑦 𝑗,𝐺 𝑗,𝐽 𝑥,𝑁 𝜑,𝑗,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑘,𝑛) 𝐴(𝑘,𝑛) 𝐹(𝑗,𝑘,𝑛) 𝐺(𝑥,𝑦,𝑘,𝑛) 𝐻(𝑥,𝑦,𝑗,𝑘,𝑛) 𝐽(𝑥,𝑦,𝑘,𝑛) 𝑁(𝑦,𝑗,𝑘,𝑛)

Proof of Theorem ennnfonelem0 Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemh.h	. . . 4 ⊢ 𝐻 = seq0(𝐺, 𝐽)
2	1	fveq1i 5630	. . 3 ⊢ (𝐻‘0) = (seq0(𝐺, 𝐽)‘0)
3		ennnfonelemh.dceq	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
4		ennnfonelemh.f	. . . . 5 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
5		ennnfonelemh.ne	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
6		ennnfonelemh.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑_pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
7		ennnfonelemh.n	. . . . 5 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
8		ennnfonelemh.j	. . . . 5 ⊢ 𝐽 = (𝑥 ∈ ℕ₀ ↦ if(𝑥 = 0, ∅, (^◡𝑁‘(𝑥 − 1))))
9	3, 4, 5, 6, 7, 8, 1	ennnfonelemj0 12988	. . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
10	3, 4, 5, 6, 7, 8, 1	ennnfonelemg 12990	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
11		0zd 9469	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
12	3, 4, 5, 6, 7, 8, 1	ennnfonelemjn 12989	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ_≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω)
13	9, 10, 11, 12	seq1cd 10703	. . 3 ⊢ (𝜑 → (seq0(𝐺, 𝐽)‘0) = (𝐽‘0))
14	2, 13	eqtrid 2274	. 2 ⊢ (𝜑 → (𝐻‘0) = (𝐽‘0))
15		0nn0 9395	. . . 4 ⊢ 0 ∈ ℕ₀
16		eqid 2229	. . . . . 6 ⊢ 0 = 0
17	16	iftruei 3608	. . . . 5 ⊢ if(0 = 0, ∅, (^◡𝑁‘(0 − 1))) = ∅
18		0ex 4211	. . . . 5 ⊢ ∅ ∈ V
19	17, 18	eqeltri 2302	. . . 4 ⊢ if(0 = 0, ∅, (^◡𝑁‘(0 − 1))) ∈ V
20		eqeq1 2236	. . . . . 6 ⊢ (𝑥 = 0 → (𝑥 = 0 ↔ 0 = 0))
21		fvoveq1 6030	. . . . . 6 ⊢ (𝑥 = 0 → (^◡𝑁‘(𝑥 − 1)) = (^◡𝑁‘(0 − 1)))
22	20, 21	ifbieq2d 3627	. . . . 5 ⊢ (𝑥 = 0 → if(𝑥 = 0, ∅, (^◡𝑁‘(𝑥 − 1))) = if(0 = 0, ∅, (^◡𝑁‘(0 − 1))))
23	22, 8	fvmptg 5712	. . . 4 ⊢ ((0 ∈ ℕ₀ ∧ if(0 = 0, ∅, (^◡𝑁‘(0 − 1))) ∈ V) → (𝐽‘0) = if(0 = 0, ∅, (^◡𝑁‘(0 − 1))))
24	15, 19, 23	mp2an 426	. . 3 ⊢ (𝐽‘0) = if(0 = 0, ∅, (^◡𝑁‘(0 − 1)))
25	24, 17	eqtri 2250	. 2 ⊢ (𝐽‘0) = ∅
26	14, 25	eqtrdi 2278	1 ⊢ (𝜑 → (𝐻‘0) = ∅)

Colors of variables: wff set class

Syntax hints: → wi 4 DECID wdc 839 = wceq 1395 ∈ wcel 2200 ≠ wne 2400 ∀wral 2508 ∃wrex 2509 {crab 2512 Vcvv 2799 ∪ cun 3195 ∅c0 3491 ifcif 3602 {csn 3666 ⟨cop 3669 ↦ cmpt 4145 suc csuc 4456 ωcom 4682 ^◡ccnv 4718 dom cdm 4719 “ cima 4722 –onto→wfo 5316 ‘cfv 5318 (class class class)co 6007 ∈ cmpo 6009 freccfrec 6542 ↑_pm cpm 6804 0cc0 8010 1c1 8011 + caddc 8013 − cmin 8328 ℕ₀cn0 9380 ℤcz 9457 seqcseq 10681

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-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-addcom 8110 ax-addass 8112 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-0id 8118 ax-rnegex 8119 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-ltadd 8126

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-pm 6806 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-inn 9122 df-n0 9381 df-z 9458 df-uz 9734 df-seqfrec 10682

This theorem is referenced by: ennnfonelem1 12994 ennnfonelemkh 12999 ennnfonelemhf1o 13000

W3C validator