Theorem nninfwlporlem 7239

Description: Lemma for nninfwlpor 7240. 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 5557	. . . . . . 7 ⊢ (𝑓 = 𝐷 → (𝑓‘𝑥) = (𝐷‘𝑥))
2	1	eqeq1d 2205	. . . . . 6 ⊢ (𝑓 = 𝐷 → ((𝑓‘𝑥) = 1o ↔ (𝐷‘𝑥) = 1o))
3	2	ralbidv 2497	. . . . 5 ⊢ (𝑓 = 𝐷 → (∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
4	3	dcbid 839	. . . 4 ⊢ (𝑓 = 𝐷 → (DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
5		nninfwlporlem.w	. . . . 5 ⊢ (𝜑 → ω ∈ WOmni)
6		omex 4629	. . . . . 6 ⊢ ω ∈ V
7		iswomnimap 7232	. . . . . 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 6500	. . . . . . . 8 ⊢ 1o ∈ 2o
11	10	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
12		0lt2o 6499	. . . . . . . 8 ⊢ ∅ ∈ 2o
13	12	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ∅ ∈ 2o)
14		2ssom 6582	. . . . . . . . 9 ⊢ 2o ⊆ ω
15		nninfwlporlem.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:ω⟶2o)
16	15	ffvelcdmda 5697	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ 2o)
17	14, 16	sselid 3181	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ ω)
18		nninfwlporlem.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌:ω⟶2o)
19	18	ffvelcdmda 5697	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ 2o)
20	14, 19	sselid 3181	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ ω)
21		nndceq 6557	. . . . . . . 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 3597	. . . . . 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 5716	. . . . 5 ⊢ (𝜑 → 𝐷:ω⟶2o)
26		2onn 6579	. . . . . . 7 ⊢ 2o ∈ ω
27	26	elexi 2775	. . . . . 6 ⊢ 2o ∈ V
28	27, 6	elmap 6736	. . . . 5 ⊢ (𝐷 ∈ (2o ↑𝑚 ω) ↔ 𝐷:ω⟶2o)
29	25, 28	sylibr 134	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (2o ↑𝑚 ω))
30	4, 9, 29	rspcdva 2873	. . 3 ⊢ (𝜑 → DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o)
31	25	ffnd 5408	. . . . 5 ⊢ (𝜑 → 𝐷 Fn ω)
32		eqidd 2197	. . . . 5 ⊢ (𝑥 = 𝑖 → 1o = 1o)
33		1onn 6578	. . . . . 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 5666	. . . 4 ⊢ (𝜑 → (𝐷 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
37	36	dcbid 839	. . 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 7238	. . 3 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ 𝐷 = (𝑖 ∈ ω ↦ 1o)))
40	39	dcbid 839	. 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 835   = wceq 1364   ∈ wcel 2167  ∀wral 2475  Vcvv 2763  ∅c0 3450  ifcif 3561   ↦ cmpt 4094  ωcom 4626  ⟶wf 5254  ‘cfv 5258  (class class class)co 5922  1_oc1o 6467  2_oc2o 6468   ↑_𝑚 cmap 6707  WOmnicwomni 7229

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1o 6474  df-2o 6475  df-map 6709  df-womni 7230

This theorem is referenced by:  nninfwlpor  7240