Theorem nninfwlporlem 7149

Description: Lemma for nninfwlpor 7150. 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 5495	. . . . . . 7 ⊢ (𝑓 = 𝐷 → (𝑓‘𝑥) = (𝐷‘𝑥))
2	1	eqeq1d 2179	. . . . . 6 ⊢ (𝑓 = 𝐷 → ((𝑓‘𝑥) = 1o ↔ (𝐷‘𝑥) = 1o))
3	2	ralbidv 2470	. . . . 5 ⊢ (𝑓 = 𝐷 → (∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
4	3	dcbid 833	. . . 4 ⊢ (𝑓 = 𝐷 → (DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
5		nninfwlporlem.w	. . . . 5 ⊢ (𝜑 → ω ∈ WOmni)
6		omex 4577	. . . . . 6 ⊢ ω ∈ V
7		iswomnimap 7142	. . . . . 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 121	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ (2o ↑𝑚 ω)DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o)
10		1lt2o 6421	. . . . . . . 8 ⊢ 1o ∈ 2o
11	10	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
12		0lt2o 6420	. . . . . . . 8 ⊢ ∅ ∈ 2o
13	12	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ∅ ∈ 2o)
14		2ssom 6503	. . . . . . . . 9 ⊢ 2o ⊆ ω
15		nninfwlporlem.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:ω⟶2o)
16	15	ffvelrnda 5631	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ 2o)
17	14, 16	sselid 3145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ ω)
18		nninfwlporlem.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌:ω⟶2o)
19	18	ffvelrnda 5631	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ 2o)
20	14, 19	sselid 3145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ ω)
21		nndceq 6478	. . . . . . . 8 ⊢ (((𝑋‘𝑖) ∈ ω ∧ (𝑌‘𝑖) ∈ ω) → DECID (𝑋‘𝑖) = (𝑌‘𝑖))
22	17, 20, 21	syl2anc 409	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → DECID (𝑋‘𝑖) = (𝑌‘𝑖))
23	11, 13, 22	ifcldcd 3561	. . . . . 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 5650	. . . . 5 ⊢ (𝜑 → 𝐷:ω⟶2o)
26		2onn 6500	. . . . . . 7 ⊢ 2o ∈ ω
27	26	elexi 2742	. . . . . 6 ⊢ 2o ∈ V
28	27, 6	elmap 6655	. . . . 5 ⊢ (𝐷 ∈ (2o ↑𝑚 ω) ↔ 𝐷:ω⟶2o)
29	25, 28	sylibr 133	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (2o ↑𝑚 ω))
30	4, 9, 29	rspcdva 2839	. . 3 ⊢ (𝜑 → DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o)
31	25	ffnd 5348	. . . . 5 ⊢ (𝜑 → 𝐷 Fn ω)
32		eqidd 2171	. . . . 5 ⊢ (𝑥 = 𝑖 → 1o = 1o)
33		1onn 6499	. . . . . 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 5600	. . . 4 ⊢ (𝜑 → (𝐷 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
37	36	dcbid 833	. . 3 ⊢ (𝜑 → (DECID 𝐷 = (𝑖 ∈ ω ↦ 1o) ↔ DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
38	30, 37	mpbird 166	. 2 ⊢ (𝜑 → DECID 𝐷 = (𝑖 ∈ ω ↦ 1o))
39	15, 18, 24	nninfwlporlemd 7148	. . 3 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ 𝐷 = (𝑖 ∈ ω ↦ 1o)))
40	39	dcbid 833	. 2 ⊢ (𝜑 → (DECID 𝑋 = 𝑌 ↔ DECID 𝐷 = (𝑖 ∈ ω ↦ 1o)))
41	38, 40	mpbird 166	1 ⊢ (𝜑 → DECID 𝑋 = 𝑌)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 829   = wceq 1348   ∈ wcel 2141  ∀wral 2448  Vcvv 2730  ∅c0 3414  ifcif 3526   ↦ cmpt 4050  ωcom 4574  ⟶wf 5194  ‘cfv 5198  (class class class)co 5853  1_oc1o 6388  2_oc2o 6389   ↑_𝑚 cmap 6626  WOmnicwomni 7139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1o 6395  df-2o 6396  df-map 6628  df-womni 7140

This theorem is referenced by:  nninfwlpor  7150