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Theorem fidcenumlemrks 6946
Description: Lemma for fidcenum 6949. Induction step for fidcenumlemrk 6947. (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 3302 . . . . 5 (𝑋 ∈ (𝐹𝐽) → 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
31, 2syl 14 . . . 4 ((𝜑𝑋 ∈ (𝐹𝐽)) → 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
4 df-suc 4368 . . . . . . 7 suc 𝐽 = (𝐽 ∪ {𝐽})
54imaeq2i 4964 . . . . . 6 (𝐹 “ suc 𝐽) = (𝐹 “ (𝐽 ∪ {𝐽}))
6 imaundi 5037 . . . . . 6 (𝐹 “ (𝐽 ∪ {𝐽})) = ((𝐹𝐽) ∪ (𝐹 “ {𝐽}))
75, 6eqtri 2198 . . . . 5 (𝐹 “ suc 𝐽) = ((𝐹𝐽) ∪ (𝐹 “ {𝐽}))
87eleq2i 2244 . . . 4 (𝑋 ∈ (𝐹 “ suc 𝐽) ↔ 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
93, 8sylibr 134 . . 3 ((𝜑𝑋 ∈ (𝐹𝐽)) → 𝑋 ∈ (𝐹 “ suc 𝐽))
109orcd 733 . 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 3606 . . . . . . . . . 10 (𝑋𝐴 → (𝑋 ∈ {(𝐹𝐽)} ↔ 𝑋 = (𝐹𝐽)))
1412, 13syl 14 . . . . . . . . 9 (𝜑 → (𝑋 ∈ {(𝐹𝐽)} ↔ 𝑋 = (𝐹𝐽)))
15 fidcenumlemr.f . . . . . . . . . . . 12 (𝜑𝐹:𝑁onto𝐴)
16 fofn 5436 . . . . . . . . . . . 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 4413 . . . . . . . . . . . . 13 (𝐽 ∈ ω → 𝐽 ∈ suc 𝐽)
2119, 20syl 14 . . . . . . . . . . . 12 (𝜑𝐽 ∈ suc 𝐽)
2218, 21sseldd 3156 . . . . . . . . . . 11 (𝜑𝐽𝑁)
23 fnsnfv 5571 . . . . . . . . . . 11 ((𝐹 Fn 𝑁𝐽𝑁) → {(𝐹𝐽)} = (𝐹 “ {𝐽}))
2417, 22, 23syl2anc 411 . . . . . . . . . 10 (𝜑 → {(𝐹𝐽)} = (𝐹 “ {𝐽}))
2524eleq2d 2247 . . . . . . . . 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 3303 . . . . . 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 733 . . 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 672 . . . . . . 7 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 ∈ (𝐹 “ {𝐽}))
37 ioran 752 . . . . . . 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 3276 . . . . . 6 (𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})) ↔ (𝑋 ∈ (𝐹𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})))
4038, 39sylnibr 677 . . . . 5 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 ∈ ((𝐹𝐽) ∪ (𝐹 “ {𝐽})))
4140, 8sylnibr 677 . . . 4 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → ¬ 𝑋 ∈ (𝐹 “ suc 𝐽))
4241olcd 734 . . 3 (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) ∧ ¬ 𝑋 = (𝐹𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
43 fof 5434 . . . . . . . 8 (𝐹:𝑁onto𝐴𝐹:𝑁𝐴)
4415, 43syl 14 . . . . . . 7 (𝜑𝐹:𝑁𝐴)
4544, 22ffvelcdmd 5648 . . . . . 6 (𝜑 → (𝐹𝐽) ∈ 𝐴)
46 fidcenumlemr.dc . . . . . 6 (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
47 eqeq1 2184 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥 = 𝑦𝑋 = 𝑦))
4847dcbid 838 . . . . . . 7 (𝑥 = 𝑋 → (DECID 𝑥 = 𝑦DECID 𝑋 = 𝑦))
49 eqeq2 2187 . . . . . . . 8 (𝑦 = (𝐹𝐽) → (𝑋 = 𝑦𝑋 = (𝐹𝐽)))
5049dcbid 838 . . . . . . 7 (𝑦 = (𝐹𝐽) → (DECID 𝑋 = 𝑦DECID 𝑋 = (𝐹𝐽)))
5148, 50rspc2va 2855 . . . . . 6 (((𝑋𝐴 ∧ (𝐹𝐽) ∈ 𝐴) ∧ ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦) → DECID 𝑋 = (𝐹𝐽))
5212, 45, 46, 51syl21anc 1237 . . . . 5 (𝜑DECID 𝑋 = (𝐹𝐽))
53 exmiddc 836 . . . . 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 798 . 2 ((𝜑 ∧ ¬ 𝑋 ∈ (𝐹𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
57 fidcenumlemrks.h . 2 (𝜑 → (𝑋 ∈ (𝐹𝐽) ∨ ¬ 𝑋 ∈ (𝐹𝐽)))
5810, 56, 57mpjaodan 798 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 708  DECID wdc 834   = wceq 1353  wcel 2148  wral 2455  cun 3127  wss 3129  {csn 3591  suc csuc 4362  ωcom 4586  cima 4626   Fn wfn 5207  wf 5208  ontowfo 5210  cfv 5212
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-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  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-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fo 5218  df-fv 5220
This theorem is referenced by:  fidcenumlemrk  6947
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