Theorem nninfwlporlem 7340

Description: Lemma for nninfwlpor 7341. 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 5626	. . . . . . 7 ⊢ (𝑓 = 𝐷 → (𝑓‘𝑥) = (𝐷‘𝑥))
2	1	eqeq1d 2238	. . . . . 6 ⊢ (𝑓 = 𝐷 → ((𝑓‘𝑥) = 1o ↔ (𝐷‘𝑥) = 1o))
3	2	ralbidv 2530	. . . . 5 ⊢ (𝑓 = 𝐷 → (∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
4	3	dcbid 843	. . . 4 ⊢ (𝑓 = 𝐷 → (DECID ∀𝑥 ∈ ω (𝑓‘𝑥) = 1o ↔ DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
5		nninfwlporlem.w	. . . . 5 ⊢ (𝜑 → ω ∈ WOmni)
6		omex 4685	. . . . . 6 ⊢ ω ∈ V
7		iswomnimap 7333	. . . . . 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 6588	. . . . . . . 8 ⊢ 1o ∈ 2o
11	10	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → 1o ∈ 2o)
12		0lt2o 6587	. . . . . . . 8 ⊢ ∅ ∈ 2o
13	12	a1i 9	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → ∅ ∈ 2o)
14		2ssom 6670	. . . . . . . . 9 ⊢ 2o ⊆ ω
15		nninfwlporlem.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:ω⟶2o)
16	15	ffvelcdmda 5770	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ 2o)
17	14, 16	sselid 3222	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑋‘𝑖) ∈ ω)
18		nninfwlporlem.y	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌:ω⟶2o)
19	18	ffvelcdmda 5770	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ 2o)
20	14, 19	sselid 3222	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ω) → (𝑌‘𝑖) ∈ ω)
21		nndceq 6645	. . . . . . . 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 3640	. . . . . 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 5789	. . . . 5 ⊢ (𝜑 → 𝐷:ω⟶2o)
26		2onn 6667	. . . . . . 7 ⊢ 2o ∈ ω
27	26	elexi 2812	. . . . . 6 ⊢ 2o ∈ V
28	27, 6	elmap 6824	. . . . 5 ⊢ (𝐷 ∈ (2o ↑𝑚 ω) ↔ 𝐷:ω⟶2o)
29	25, 28	sylibr 134	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (2o ↑𝑚 ω))
30	4, 9, 29	rspcdva 2912	. . 3 ⊢ (𝜑 → DECID ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o)
31	25	ffnd 5474	. . . . 5 ⊢ (𝜑 → 𝐷 Fn ω)
32		eqidd 2230	. . . . 5 ⊢ (𝑥 = 𝑖 → 1o = 1o)
33		1onn 6666	. . . . . 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 5739	. . . 4 ⊢ (𝜑 → (𝐷 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝐷‘𝑥) = 1o))
37	36	dcbid 843	. . 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 7339	. . 3 ⊢ (𝜑 → (𝑋 = 𝑌 ↔ 𝐷 = (𝑖 ∈ ω ↦ 1o)))
40	39	dcbid 843	. 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 839   = wceq 1395   ∈ wcel 2200  ∀wral 2508  Vcvv 2799  ∅c0 3491  ifcif 3602   ↦ cmpt 4145  ωcom 4682  ⟶wf 5314  ‘cfv 5318  (class class class)co 6001  1_oc1o 6555  2_oc2o 6556   ↑_𝑚 cmap 6795  WOmnicwomni 7330

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-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

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-ral 2513  df-rex 2514  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-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  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-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1o 6562  df-2o 6563  df-map 6797  df-womni 7331

This theorem is referenced by:  nninfwlpor  7341