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Theorem nninfwlporlem 7173
Description: Lemma for nninfwlpor 7174. 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 5516 . . . . . . 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 4594 . . . . . 6 ω ∈ V
7 iswomnimap 7166 . . . . . 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 6445 . . . . . . . 8 1o ∈ 2o
1110a1i 9 . . . . . . 7 ((𝜑𝑖 ∈ ω) → 1o ∈ 2o)
12 0lt2o 6444 . . . . . . . 8 ∅ ∈ 2o
1312a1i 9 . . . . . . 7 ((𝜑𝑖 ∈ ω) → ∅ ∈ 2o)
14 2ssom 6527 . . . . . . . . 9 2o ⊆ ω
15 nninfwlporlem.x . . . . . . . . . 10 (𝜑𝑋:ω⟶2o)
1615ffvelcdmda 5653 . . . . . . . . 9 ((𝜑𝑖 ∈ ω) → (𝑋𝑖) ∈ 2o)
1714, 16sselid 3155 . . . . . . . 8 ((𝜑𝑖 ∈ ω) → (𝑋𝑖) ∈ ω)
18 nninfwlporlem.y . . . . . . . . . 10 (𝜑𝑌:ω⟶2o)
1918ffvelcdmda 5653 . . . . . . . . 9 ((𝜑𝑖 ∈ ω) → (𝑌𝑖) ∈ 2o)
2014, 19sselid 3155 . . . . . . . 8 ((𝜑𝑖 ∈ ω) → (𝑌𝑖) ∈ ω)
21 nndceq 6502 . . . . . . . 8 (((𝑋𝑖) ∈ ω ∧ (𝑌𝑖) ∈ ω) → DECID (𝑋𝑖) = (𝑌𝑖))
2217, 20, 21syl2anc 411 . . . . . . 7 ((𝜑𝑖 ∈ ω) → DECID (𝑋𝑖) = (𝑌𝑖))
2311, 13, 22ifcldcd 3572 . . . . . 6 ((𝜑𝑖 ∈ ω) → if((𝑋𝑖) = (𝑌𝑖), 1o, ∅) ∈ 2o)
24 nninfwlporlem.d . . . . . 6 𝐷 = (𝑖 ∈ ω ↦ if((𝑋𝑖) = (𝑌𝑖), 1o, ∅))
2523, 24fmptd 5672 . . . . 5 (𝜑𝐷:ω⟶2o)
26 2onn 6524 . . . . . . 7 2o ∈ ω
2726elexi 2751 . . . . . 6 2o ∈ V
2827, 6elmap 6679 . . . . 5 (𝐷 ∈ (2o𝑚 ω) ↔ 𝐷:ω⟶2o)
2925, 28sylibr 134 . . . 4 (𝜑𝐷 ∈ (2o𝑚 ω))
304, 9, 29rspcdva 2848 . . 3 (𝜑DECID𝑥 ∈ ω (𝐷𝑥) = 1o)
3125ffnd 5368 . . . . 5 (𝜑𝐷 Fn ω)
32 eqidd 2178 . . . . 5 (𝑥 = 𝑖 → 1o = 1o)
33 1onn 6523 . . . . . 6 1o ∈ ω
3433a1i 9 . . . . 5 ((𝜑𝑥 ∈ ω) → 1o ∈ ω)
3533a1i 9 . . . . 5 ((𝜑𝑖 ∈ ω) → 1o ∈ ω)
3631, 32, 34, 35fnmptfvd 5622 . . . 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 7172 . . 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 2739  c0 3424  ifcif 3536  cmpt 4066  ωcom 4591  wf 5214  cfv 5218  (class class class)co 5877  1oc1o 6412  2oc2o 6413  𝑚 cmap 6650  WOmnicwomni 7163
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 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589
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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1o 6419  df-2o 6420  df-map 6652  df-womni 7164
This theorem is referenced by:  nninfwlpor  7174
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