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Theorem fidcenumlemrks 6841
 Description: Lemma for fidcenum 6844. Induction step for fidcenumlemrk 6842. (Contributed by Jim Kingdon, 20-Oct-2022.)
Hypotheses
Ref Expression
fidcenumlemr.dc (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
fidcenumlemr.f (𝜑𝐹:𝑁onto𝐴)
fidcenumlemrks.j (𝜑𝐽 ∈ ω)
fidcenumlemrks.jn (𝜑 → suc 𝐽𝑁)
fidcenumlemrks.h (𝜑 → (𝑋 ∈ (𝐹𝐽) ∨ ¬ 𝑋 ∈ (𝐹𝐽)))
fidcenumlemrks.x (𝜑𝑋𝐴)
Assertion
Ref Expression
fidcenumlemrks (𝜑 → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐹   𝑦,𝐽   𝑥,𝑋,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐹(𝑥)   𝐽(𝑥)   𝑁(𝑥,𝑦)

Proof of Theorem fidcenumlemrks
StepHypRef Expression
1 simpr 109 . . . . 5 ((𝜑𝑋 ∈ (𝐹𝐽)) → 𝑋 ∈ (𝐹𝐽))
2 elun1 3243 . . . . 5 (𝑋 ∈ (𝐹𝐽) → 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
31, 2syl 14 . . . 4 ((𝜑𝑋 ∈ (𝐹𝐽)) → 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
4 df-suc 4293 . . . . . . 7 suc 𝐽 = (𝐽 ∪ {𝐽})
54imaeq2i 4879 . . . . . 6 (𝐹 “ suc 𝐽) = (𝐹 “ (𝐽 ∪ {𝐽}))
6 imaundi 4951 . . . . . 6 (𝐹 “ (𝐽 ∪ {𝐽})) = ((𝐹𝐽) ∪ (𝐹 “ {𝐽}))
75, 6eqtri 2160 . . . . 5 (𝐹 “ suc 𝐽) = ((𝐹𝐽) ∪ (𝐹 “ {𝐽}))
87eleq2i 2206 . . . 4 (𝑋 ∈ (𝐹 “ suc 𝐽) ↔ 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
93, 8sylibr 133 . . 3 ((𝜑𝑋 ∈ (𝐹𝐽)) → 𝑋 ∈ (𝐹 “ suc 𝐽))
109orcd 722 . 2 ((𝜑𝑋 ∈ (𝐹𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
11 simpr 109 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → 𝑋 = (𝐹𝐽))
12 fidcenumlemrks.x . . . . . . . . . 10 (𝜑𝑋𝐴)
13 elsng 3542 . . . . . . . . . 10 (𝑋𝐴 → (𝑋 ∈ {(𝐹𝐽)} ↔ 𝑋 = (𝐹𝐽)))
1412, 13syl 14 . . . . . . . . 9 (𝜑 → (𝑋 ∈ {(𝐹𝐽)} ↔ 𝑋 = (𝐹𝐽)))
15 fidcenumlemr.f . . . . . . . . . . . 12 (𝜑𝐹:𝑁onto𝐴)
16 fofn 5347 . . . . . . . . . . . 12 (𝐹:𝑁onto𝐴𝐹 Fn 𝑁)
1715, 16syl 14 . . . . . . . . . . 11 (𝜑𝐹 Fn 𝑁)
18 fidcenumlemrks.jn . . . . . . . . . . . 12 (𝜑 → suc 𝐽𝑁)
19 fidcenumlemrks.j . . . . . . . . . . . . 13 (𝜑𝐽 ∈ ω)
20 sucidg 4338 . . . . . . . . . . . . 13 (𝐽 ∈ ω → 𝐽 ∈ suc 𝐽)
2119, 20syl 14 . . . . . . . . . . . 12 (𝜑𝐽 ∈ suc 𝐽)
2218, 21sseldd 3098 . . . . . . . . . . 11 (𝜑𝐽𝑁)
23 fnsnfv 5480 . . . . . . . . . . 11 ((𝐹 Fn 𝑁𝐽𝑁) → {(𝐹𝐽)} = (𝐹 “ {𝐽}))
2417, 22, 23syl2anc 408 . . . . . . . . . 10 (𝜑 → {(𝐹𝐽)} = (𝐹 “ {𝐽}))
2524eleq2d 2209 . . . . . . . . 9 (𝜑 → (𝑋 ∈ {(𝐹𝐽)} ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
2614, 25bitr3d 189 . . . . . . . 8 (𝜑 → (𝑋 = (𝐹𝐽) ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
2726ad2antrr 479 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → (𝑋 = (𝐹𝐽) ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
2811, 27mpbid 146 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → 𝑋 ∈ (𝐹 “ {𝐽}))
29 elun2 3244 . . . . . 6 (𝑋 ∈ (𝐹 “ {𝐽}) → 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
3028, 29syl 14 . . . . 5 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
3130, 8sylibr 133 . . . 4 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → 𝑋 ∈ (𝐹 “ suc 𝐽))
3231orcd 722 . . 3 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
33 simplr 519 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 ∈ (𝐹𝐽))
34 simpr 109 . . . . . . . 8 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 = (𝐹𝐽))
3526ad2antrr 479 . . . . . . . 8 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → (𝑋 = (𝐹𝐽) ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
3634, 35mtbid 661 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 ∈ (𝐹 “ {𝐽}))
37 ioran 741 . . . . . . 7 (¬ (𝑋 ∈ (𝐹𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})) ↔ (¬ 𝑋 ∈ (𝐹𝐽) ∧ ¬ 𝑋 ∈ (𝐹 “ {𝐽})))
3833, 36, 37sylanbrc 413 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ (𝑋 ∈ (𝐹𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})))
39 elun 3217 . . . . . 6 (𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})) ↔ (𝑋 ∈ (𝐹𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})))
4038, 39sylnibr 666 . . . . 5 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
4140, 8sylnibr 666 . . . 4 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 ∈ (𝐹 “ suc 𝐽))
4241olcd 723 . . 3 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
43 fof 5345 . . . . . . . 8 (𝐹:𝑁onto𝐴𝐹:𝑁𝐴)
4415, 43syl 14 . . . . . . 7 (𝜑𝐹:𝑁𝐴)
4544, 22ffvelrnd 5556 . . . . . 6 (𝜑 → (𝐹𝐽) ∈ 𝐴)
46 fidcenumlemr.dc . . . . . 6 (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
47 eqeq1 2146 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
4847dcbid 823 . . . . . . 7 (𝑥 = 𝑋 → (DECID 𝑥 = 𝑦DECID 𝑋 = 𝑦))
49 eqeq2 2149 . . . . . . . 8 (𝑦 = (𝐹𝐽) → (𝑋 = 𝑦𝑋 = (𝐹𝐽)))
5049dcbid 823 . . . . . . 7 (𝑦 = (𝐹𝐽) → (DECID 𝑋 = 𝑦DECID 𝑋 = (𝐹𝐽)))
5148, 50rspc2va 2803 . . . . . 6 (((𝑋𝐴 ∧ (𝐹𝐽) ∈ 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦) → DECID 𝑋 = (𝐹𝐽))
5212, 45, 46, 51syl21anc 1215 . . . . 5 (𝜑DECID 𝑋 = (𝐹𝐽))
53 exmiddc 821 . . . . 5 (DECID 𝑋 = (𝐹𝐽) → (𝑋 = (𝐹𝐽) ∨ ¬ 𝑋 = (𝐹𝐽)))
5452, 53syl 14 . . . 4 (𝜑 → (𝑋 = (𝐹𝐽) ∨ ¬ 𝑋 = (𝐹𝐽)))
5554adantr 274 . . 3 ((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) → (𝑋 = (𝐹𝐽) ∨ ¬ 𝑋 = (𝐹𝐽)))
5632, 42, 55mpjaodan 787 . 2 ((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
57 fidcenumlemrks.h . 2 (𝜑 → (𝑋 ∈ (𝐹𝐽) ∨ ¬ 𝑋 ∈ (𝐹𝐽)))
5810, 56, 57mpjaodan 787 1 (𝜑 → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697  DECID wdc 819   = wceq 1331   ∈ wcel 1480  ∀wral 2416   ∪ cun 3069   ⊆ wss 3071  {csn 3527  suc csuc 4287  ωcom 4504   “ cima 4542   Fn wfn 5118  ⟶wf 5119  –onto→wfo 5121  ‘cfv 5123 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131 This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131 This theorem is referenced by:  fidcenumlemrk  6842
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