Theorem nninfwlporlem 7455

Description: Lemma for nninfwlpor 7456. 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 5660	. . . . . . 7 ⊢ (𝑓 = 𝐷 → (𝑓‘𝑥) = (𝐷‘𝑥))
2	1	eqeq1d 2241	. . . . . 6 ⊢ (𝑓 = 𝐷 → ((𝑓‘𝑥) = 1o ↔ (𝐷‘𝑥) = 1o))
3	2	ralbidv 2542	. . . . 5 ⊢ (𝑓 = 𝐷 → (∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
4	3	dcbid 846	. . . 4 ⊢ (𝑓 = 𝐷 → (DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
5		nninfwlporlem.w	. . . . 5 ⊢ (𝜑 → ω ∈ WOmni)
6		omex 4706	. . . . . 6 ⊢ ω ∈ V
7		iswomnimap 7448	. . . . . 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 6666	. . . . . . . 8 ⊢ 1o ∈ 2o
11	10	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
12		0lt2o 6665	. . . . . . . 8 ⊢ ∅ ∈ 2o
13	12	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ∅ ∈ 2o)
14		2ssom 6748	. . . . . . . . 9 ⊢ 2o ⊆ ω
15		nninfwlporlem.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:ω⟶2o)
16	15	ffvelcdmda 5803	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ 2o)
17	14, 16	sselid 3235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ ω)
18		nninfwlporlem.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌:ω⟶2o)
19	18	ffvelcdmda 5803	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ 2o)
20	14, 19	sselid 3235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ ω)
21		nndceq 6723	. . . . . . . 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 3656	. . . . . 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 5822	. . . . 5 ⊢ (𝜑 → 𝐷:ω⟶2o)
26		2onn 6745	. . . . . . 7 ⊢ 2o ∈ ω
27	26	elexi 2825	. . . . . 6 ⊢ 2o ∈ V
28	27, 6	elmap 6902	. . . . 5 ⊢ (𝐷 ∈ (2o ↑𝑚 ω) ↔ 𝐷:ω⟶2o)
29	25, 28	sylibr 134	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (2o ↑𝑚 ω))
30	4, 9, 29	rspcdva 2925	. . 3 ⊢ (𝜑 → DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o)
31	25	ffnd 5500	. . . . 5 ⊢ (𝜑 → 𝐷 Fn ω)
32		eqidd 2233	. . . . 5 ⊢ (𝑥 = 𝑖 → 1o = 1o)
33		1onn 6744	. . . . . 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 5773	. . . 4 ⊢ (𝜑 → (𝐷 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
37	36	dcbid 846	. . 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 7454	. . 3 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ 𝐷 = (𝑖 ∈ ω ↦ 1o)))
40	39	dcbid 846	. 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 842   = wceq 1398   ∈ wcel 2203  ∀wral 2520  Vcvv 2812  ∅c0 3505  ifcif 3616   ↦ cmpt 4164  ωcom 4703  ⟶wf 5339  ‘cfv 5343  (class class class)co 6041  1_oc1o 6631  2_oc2o 6632   ↑_𝑚 cmap 6873  WOmnicwomni 7445

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4221  ax-nul 4229  ax-pow 4279  ax-pr 4314  ax-un 4545  ax-setind 4650  ax-iinf 4701

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3506  df-if 3617  df-pw 3667  df-sn 3688  df-pr 3689  df-op 3691  df-uni 3908  df-int 3943  df-br 4103  df-opab 4165  df-mpt 4166  df-tr 4202  df-id 4405  df-iord 4478  df-on 4480  df-suc 4483  df-iom 4704  df-xp 4746  df-rel 4747  df-cnv 4748  df-co 4749  df-dm 4750  df-rn 4751  df-res 4752  df-ima 4753  df-iota 5303  df-fun 5345  df-fn 5346  df-f 5347  df-fv 5351  df-ov 6044  df-oprab 6045  df-mpo 6046  df-1o 6638  df-2o 6639  df-map 6875  df-womni 7446

This theorem is referenced by:  nninfwlpor  7456