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Theorem nninfwlporlem 7171
Description: Lemma for nninfwlpor 7172. The result. (Contributed by Jim Kingdon, 7-Dec-2024.)
Hypotheses
Ref Expression
nninfwlporlem.x (𝜑𝑋:ω⟶2o)
nninfwlporlem.y (𝜑𝑌:ω⟶2o)
nninfwlporlem.d 𝐷 = (𝑖 ∈ ω ↦ if((𝑋𝑖) = (𝑌𝑖), 1o, ∅))
nninfwlporlem.w (𝜑 → ω ∈ WOmni)
Assertion
Ref Expression
nninfwlporlem (𝜑DECID 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.
StepHypRef Expression
1 fveq1 5515 . . . . . . 7 (𝑓 = 𝐷 → (𝑓𝑥) = (𝐷𝑥))
21eqeq1d 2186 . . . . . 6 (𝑓 = 𝐷 → ((𝑓𝑥) = 1o ↔ (𝐷𝑥) = 1o))
32ralbidv 2477 . . . . 5 (𝑓 = 𝐷 → (∀𝑥 ∈ ω (𝑓𝑥) = 1o ↔ ∀𝑥 ∈ ω (𝐷𝑥) = 1o))
43dcbid 838 . . . 4 (𝑓 = 𝐷 → (DECID𝑥 ∈ ω (𝑓𝑥) = 1oDECID𝑥 ∈ ω (𝐷𝑥) = 1o))
5 nninfwlporlem.w . . . . 5 (𝜑 → ω ∈ WOmni)
6 omex 4593 . . . . . 6 ω ∈ V
7 iswomnimap 7164 . . . . . 6 (ω ∈ V → (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o𝑚 ω)DECID𝑥 ∈ ω (𝑓𝑥) = 1o))
86, 7ax-mp 5 . . . . 5 (ω ∈ WOmni ↔ ∀𝑓 ∈ (2o𝑚 ω)DECID𝑥 ∈ ω (𝑓𝑥) = 1o)
95, 8sylib 122 . . . 4 (𝜑 → ∀𝑓 ∈ (2o𝑚 ω)DECID𝑥 ∈ ω (𝑓𝑥) = 1o)
10 1lt2o 6443 . . . . . . . 8 1o ∈ 2o
1110a1i 9 . . . . . . 7 ((𝜑𝑖 ∈ ω) → 1o ∈ 2o)
12 0lt2o 6442 . . . . . . . 8 ∅ ∈ 2o
1312a1i 9 . . . . . . 7 ((𝜑𝑖 ∈ ω) → ∅ ∈ 2o)
14 2ssom 6525 . . . . . . . . 9 2o ⊆ ω
15 nninfwlporlem.x . . . . . . . . . 10 (𝜑𝑋:ω⟶2o)
1615ffvelcdmda 5652 . . . . . . . . 9 ((𝜑𝑖 ∈ ω) → (𝑋𝑖) ∈ 2o)
1714, 16sselid 3154 . . . . . . . 8 ((𝜑𝑖 ∈ ω) → (𝑋𝑖) ∈ ω)
18 nninfwlporlem.y . . . . . . . . . 10 (𝜑𝑌:ω⟶2o)
1918ffvelcdmda 5652 . . . . . . . . 9 ((𝜑𝑖 ∈ ω) → (𝑌𝑖) ∈ 2o)
2014, 19sselid 3154 . . . . . . . 8 ((𝜑𝑖 ∈ ω) → (𝑌𝑖) ∈ ω)
21 nndceq 6500 . . . . . . . 8 (((𝑋𝑖) ∈ ω ∧ (𝑌𝑖) ∈ ω) → DECID (𝑋𝑖) = (𝑌𝑖))
2217, 20, 21syl2anc 411 . . . . . . 7 ((𝜑𝑖 ∈ ω) → DECID (𝑋𝑖) = (𝑌𝑖))
2311, 13, 22ifcldcd 3571 . . . . . 6 ((𝜑𝑖 ∈ ω) → if((𝑋𝑖) = (𝑌𝑖), 1o, ∅) ∈ 2o)
24 nninfwlporlem.d . . . . . 6 𝐷 = (𝑖 ∈ ω ↦ if((𝑋𝑖) = (𝑌𝑖), 1o, ∅))
2523, 24fmptd 5671 . . . . 5 (𝜑𝐷:ω⟶2o)
26 2onn 6522 . . . . . . 7 2o ∈ ω
2726elexi 2750 . . . . . 6 2o ∈ V
2827, 6elmap 6677 . . . . 5 (𝐷 ∈ (2o𝑚 ω) ↔ 𝐷:ω⟶2o)
2925, 28sylibr 134 . . . 4 (𝜑𝐷 ∈ (2o𝑚 ω))
304, 9, 29rspcdva 2847 . . 3 (𝜑DECID𝑥 ∈ ω (𝐷𝑥) = 1o)
3125ffnd 5367 . . . . 5 (𝜑𝐷 Fn ω)
32 eqidd 2178 . . . . 5 (𝑥 = 𝑖 → 1o = 1o)
33 1onn 6521 . . . . . 6 1o ∈ ω
3433a1i 9 . . . . 5 ((𝜑𝑥 ∈ ω) → 1o ∈ ω)
3533a1i 9 . . . . 5 ((𝜑𝑖 ∈ ω) → 1o ∈ ω)
3631, 32, 34, 35fnmptfvd 5621 . . . 4 (𝜑 → (𝐷 = (𝑖 ∈ ω ↦ 1o) ↔ ∀𝑥 ∈ ω (𝐷𝑥) = 1o))
3736dcbid 838 . . 3 (𝜑 → (DECID 𝐷 = (𝑖 ∈ ω ↦ 1o) ↔ DECID𝑥 ∈ ω (𝐷𝑥) = 1o))
3830, 37mpbird 167 . 2 (𝜑DECID 𝐷 = (𝑖 ∈ ω ↦ 1o))
3915, 18, 24nninfwlporlemd 7170 . . 3 (𝜑 → (𝑋 = 𝑌𝐷 = (𝑖 ∈ ω ↦ 1o)))
4039dcbid 838 . 2 (𝜑 → (DECID 𝑋 = 𝑌DECID 𝐷 = (𝑖 ∈ ω ↦ 1o)))
4138, 40mpbird 167 1 (𝜑DECID 𝑋 = 𝑌)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  DECID wdc 834   = wceq 1353  wcel 2148  wral 2455  Vcvv 2738  c0 3423  ifcif 3535  cmpt 4065  ωcom 4590  wf 5213  cfv 5217  (class class class)co 5875  1oc1o 6410  2oc2o 6411  𝑚 cmap 6648  WOmnicwomni 7161
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1o 6417  df-2o 6418  df-map 6650  df-womni 7162
This theorem is referenced by:  nninfwlpor  7172
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