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Theorem fidcenumlemrks 6970
Description: Lemma for fidcenum 6973. Induction step for fidcenumlemrk 6971. (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 110 . . . . 5 ((𝜑𝑋 ∈ (𝐹𝐽)) → 𝑋 ∈ (𝐹𝐽))
2 elun1 3317 . . . . 5 (𝑋 ∈ (𝐹𝐽) → 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
31, 2syl 14 . . . 4 ((𝜑𝑋 ∈ (𝐹𝐽)) → 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
4 df-suc 4386 . . . . . . 7 suc 𝐽 = (𝐽 ∪ {𝐽})
54imaeq2i 4983 . . . . . 6 (𝐹 “ suc 𝐽) = (𝐹 “ (𝐽 ∪ {𝐽}))
6 imaundi 5056 . . . . . 6 (𝐹 “ (𝐽 ∪ {𝐽})) = ((𝐹𝐽) ∪ (𝐹 “ {𝐽}))
75, 6eqtri 2210 . . . . 5 (𝐹 “ suc 𝐽) = ((𝐹𝐽) ∪ (𝐹 “ {𝐽}))
87eleq2i 2256 . . . 4 (𝑋 ∈ (𝐹 “ suc 𝐽) ↔ 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
93, 8sylibr 134 . . 3 ((𝜑𝑋 ∈ (𝐹𝐽)) → 𝑋 ∈ (𝐹 “ suc 𝐽))
109orcd 734 . 2 ((𝜑𝑋 ∈ (𝐹𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
11 simpr 110 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → 𝑋 = (𝐹𝐽))
12 fidcenumlemrks.x . . . . . . . . . 10 (𝜑𝑋𝐴)
13 elsng 3622 . . . . . . . . . 10 (𝑋𝐴 → (𝑋 ∈ {(𝐹𝐽)} ↔ 𝑋 = (𝐹𝐽)))
1412, 13syl 14 . . . . . . . . 9 (𝜑 → (𝑋 ∈ {(𝐹𝐽)} ↔ 𝑋 = (𝐹𝐽)))
15 fidcenumlemr.f . . . . . . . . . . . 12 (𝜑𝐹:𝑁onto𝐴)
16 fofn 5455 . . . . . . . . . . . 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 4431 . . . . . . . . . . . . 13 (𝐽 ∈ ω → 𝐽 ∈ suc 𝐽)
2119, 20syl 14 . . . . . . . . . . . 12 (𝜑𝐽 ∈ suc 𝐽)
2218, 21sseldd 3171 . . . . . . . . . . 11 (𝜑𝐽𝑁)
23 fnsnfv 5591 . . . . . . . . . . 11 ((𝐹 Fn 𝑁𝐽𝑁) → {(𝐹𝐽)} = (𝐹 “ {𝐽}))
2417, 22, 23syl2anc 411 . . . . . . . . . 10 (𝜑 → {(𝐹𝐽)} = (𝐹 “ {𝐽}))
2524eleq2d 2259 . . . . . . . . 9 (𝜑 → (𝑋 ∈ {(𝐹𝐽)} ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
2614, 25bitr3d 190 . . . . . . . 8 (𝜑 → (𝑋 = (𝐹𝐽) ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
2726ad2antrr 488 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → (𝑋 = (𝐹𝐽) ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
2811, 27mpbid 147 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → 𝑋 ∈ (𝐹 “ {𝐽}))
29 elun2 3318 . . . . . 6 (𝑋 ∈ (𝐹 “ {𝐽}) → 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
3028, 29syl 14 . . . . 5 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
3130, 8sylibr 134 . . . 4 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → 𝑋 ∈ (𝐹 “ suc 𝐽))
3231orcd 734 . . 3 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ 𝑋 = (𝐹𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
33 simplr 528 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 ∈ (𝐹𝐽))
34 simpr 110 . . . . . . . 8 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 = (𝐹𝐽))
3526ad2antrr 488 . . . . . . . 8 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → (𝑋 = (𝐹𝐽) ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
3634, 35mtbid 673 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 ∈ (𝐹 “ {𝐽}))
37 ioran 753 . . . . . . 7 (¬ (𝑋 ∈ (𝐹𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})) ↔ (¬ 𝑋 ∈ (𝐹𝐽) ∧ ¬ 𝑋 ∈ (𝐹 “ {𝐽})))
3833, 36, 37sylanbrc 417 . . . . . 6 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ (𝑋 ∈ (𝐹𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})))
39 elun 3291 . . . . . 6 (𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})) ↔ (𝑋 ∈ (𝐹𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})))
4038, 39sylnibr 678 . . . . 5 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
4140, 8sylnibr 678 . . . 4 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 ∈ (𝐹 “ suc 𝐽))
4241olcd 735 . . 3 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
43 fof 5453 . . . . . . . 8 (𝐹:𝑁onto𝐴𝐹:𝑁𝐴)
4415, 43syl 14 . . . . . . 7 (𝜑𝐹:𝑁𝐴)
4544, 22ffvelcdmd 5668 . . . . . 6 (𝜑 → (𝐹𝐽) ∈ 𝐴)
46 fidcenumlemr.dc . . . . . 6 (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
47 eqeq1 2196 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
4847dcbid 839 . . . . . . 7 (𝑥 = 𝑋 → (DECID 𝑥 = 𝑦DECID 𝑋 = 𝑦))
49 eqeq2 2199 . . . . . . . 8 (𝑦 = (𝐹𝐽) → (𝑋 = 𝑦𝑋 = (𝐹𝐽)))
5049dcbid 839 . . . . . . 7 (𝑦 = (𝐹𝐽) → (DECID 𝑋 = 𝑦DECID 𝑋 = (𝐹𝐽)))
5148, 50rspc2va 2870 . . . . . 6 (((𝑋𝐴 ∧ (𝐹𝐽) ∈ 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦) → DECID 𝑋 = (𝐹𝐽))
5212, 45, 46, 51syl21anc 1248 . . . . 5 (𝜑DECID 𝑋 = (𝐹𝐽))
53 exmiddc 837 . . . . 5 (DECID 𝑋 = (𝐹𝐽) → (𝑋 = (𝐹𝐽) ∨ ¬ 𝑋 = (𝐹𝐽)))
5452, 53syl 14 . . . 4 (𝜑 → (𝑋 = (𝐹𝐽) ∨ ¬ 𝑋 = (𝐹𝐽)))
5554adantr 276 . . 3 ((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) → (𝑋 = (𝐹𝐽) ∨ ¬ 𝑋 = (𝐹𝐽)))
5632, 42, 55mpjaodan 799 . 2 ((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
57 fidcenumlemrks.h . 2 (𝜑 → (𝑋 ∈ (𝐹𝐽) ∨ ¬ 𝑋 ∈ (𝐹𝐽)))
5810, 56, 57mpjaodan 799 1 (𝜑 → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  DECID wdc 835   = wceq 1364  wcel 2160  wral 2468  cun 3142  wss 3144  {csn 3607  suc csuc 4380  ωcom 4604  cima 4644   Fn wfn 5226  wf 5227  ontowfo 5229  cfv 5231
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-suc 4386  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-fo 5237  df-fv 5239
This theorem is referenced by:  fidcenumlemrk  6971
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