Theorem nninfwlporlem 7372

Description: Lemma for nninfwlpor 7373. The result. (Contributed by Jim Kingdon, 7-Dec-2024.)

Hypotheses

Ref	Expression
nninfwlporlem.x	⊢ (𝜑 → 𝑋:ω⟶2o)
nninfwlporlem.y	⊢ (𝜑 → 𝑌:ω⟶2o)
nninfwlporlem.d	⊢ 𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))
nninfwlporlem.w	⊢ (𝜑 → ω ∈ WOmni)

Assertion

Ref	Expression
nninfwlporlem	⊢ (𝜑 → DECID 𝑋 = 𝑌)

Distinct variable groups:   𝐷,𝑖   𝜑,𝑖   𝑖,𝑋   𝑖,𝑌

Proof of Theorem nninfwlporlem

Dummy variables 𝑓 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq1 5638	. . . . . . 7 ⊢ (𝑓 = 𝐷 → (𝑓‘𝑥) = (𝐷‘𝑥))
2	1	eqeq1d 2240	. . . . . 6 ⊢ (𝑓 = 𝐷 → ((𝑓‘𝑥) = 1o ↔ (𝐷‘𝑥) = 1o))
3	2	ralbidv 2532	. . . . 5 ⊢ (𝑓 = 𝐷 → (∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
4	3	dcbid 845	. . . 4 ⊢ (𝑓 = 𝐷 → (DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
5		nninfwlporlem.w	. . . . 5 ⊢ (𝜑 → ω ∈ WOmni)
6		omex 4691	. . . . . 6 ⊢ ω ∈ V
7		iswomnimap 7365	. . . . . 6 ⊢ (ω ∈ V → (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o))
8	6, 7	ax-mp 5	. . . . 5 ⊢ (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o)
9	5, 8	sylib 122	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o)
10		1lt2o 6610	. . . . . . . 8 ⊢ 1o ∈ 2o
11	10	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
12		0lt2o 6609	. . . . . . . 8 ⊢ ∅ ∈ 2o
13	12	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ∅ ∈ 2o)
14		2ssom 6692	. . . . . . . . 9 ⊢ 2o ⊆ ω
15		nninfwlporlem.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:ω⟶2o)
16	15	ffvelcdmda 5782	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ 2o)
17	14, 16	sselid 3225	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ ω)
18		nninfwlporlem.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌:ω⟶2o)
19	18	ffvelcdmda 5782	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ 2o)
20	14, 19	sselid 3225	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ ω)
21		nndceq 6667	. . . . . . . 8 ⊢ (((𝑋‘𝑖) ∈ ω ∧ (𝑌‘𝑖) ∈ ω) → DECID (𝑋‘𝑖) = (𝑌‘𝑖))
22	17, 20, 21	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → DECID (𝑋‘𝑖) = (𝑌‘𝑖))
23	11, 13, 22	ifcldcd 3643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅) ∈ 2o)
24		nninfwlporlem.d	. . . . . 6 ⊢ 𝐷 = (𝑖 ∈ ω ↦ if((𝑋‘𝑖) = (𝑌‘𝑖), 1o, ∅))
25	23, 24	fmptd 5801	. . . . 5 ⊢ (𝜑 → 𝐷:ω⟶2o)
26		2onn 6689	. . . . . . 7 ⊢ 2o ∈ ω
27	26	elexi 2815	. . . . . 6 ⊢ 2o ∈ V
28	27, 6	elmap 6846	. . . . 5 ⊢ (𝐷 ∈ (2o ↑𝑚 ω) ↔ 𝐷:ω⟶2o)
29	25, 28	sylibr 134	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (2o ↑𝑚 ω))
30	4, 9, 29	rspcdva 2915	. . 3 ⊢ (𝜑 → DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o)
31	25	ffnd 5483	. . . . 5 ⊢ (𝜑 → 𝐷 Fn ω)
32		eqidd 2232	. . . . 5 ⊢ (𝑥 = 𝑖 → 1o = 1o)
33		1onn 6688	. . . . . 6 ⊢ 1o ∈ ω
34	33	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ω) → 1o ∈ ω)
35	33	a1i 9	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ ω)
36	31, 32, 34, 35	fnmptfvd 5751	. . . 4 ⊢ (𝜑 → (𝐷 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
37	36	dcbid 845	. . 3 ⊢ (𝜑 → (DECID 𝐷 = (𝑖 ∈ ω ↦ 1o) ↔ DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
38	30, 37	mpbird 167	. 2 ⊢ (𝜑 → DECID 𝐷 = (𝑖 ∈ ω ↦ 1o))
39	15, 18, 24	nninfwlporlemd 7371	. . 3 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ 𝐷 = (𝑖 ∈ ω ↦ 1o)))
40	39	dcbid 845	. 2 ⊢ (𝜑 → (DECID 𝑋 = 𝑌 ↔ DECID 𝐷 = (𝑖 ∈ ω ↦ 1o)))
41	38, 40	mpbird 167	1 ⊢ (𝜑 → DECID 𝑋 = 𝑌)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510  Vcvv 2802  ∅c0 3494  ifcif 3605   ↦ cmpt 4150  ωcom 4688  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018  1_oc1o 6575  2_oc2o 6576   ↑_𝑚 cmap 6817  WOmnicwomni 7362

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1o 6582  df-2o 6583  df-map 6819  df-womni 7363

This theorem is referenced by:  nninfwlpor  7373