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 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
fidcenumlemr.f	⊢ (𝜑 → 𝐹:𝑁–onto→𝐴)
fidcenumlemrks.j	⊢ (𝜑 → 𝐽 ∈ ω)
fidcenumlemrks.jn	⊢ (𝜑 → suc 𝐽 ⊆ 𝑁)
fidcenumlemrks.h	⊢ (𝜑 → (𝑋 ∈ (𝐹 “ 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐽)))
fidcenumlemrks.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)

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

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐹 “ 𝐽)) → 𝑋 ∈ (𝐹 “ 𝐽))
2		elun1 3317	. . . . 5 ⊢ (𝑋 ∈ (𝐹 “ 𝐽) → 𝑋 ∈ ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽})))
3	1, 2	syl 14	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐹 “ 𝐽)) → 𝑋 ∈ ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽})))
4		df-suc 4386	. . . . . . 7 ⊢ suc 𝐽 = (𝐽 ∪ {𝐽})
5	4	imaeq2i 4983	. . . . . 6 ⊢ (𝐹 “ suc 𝐽) = (𝐹 “ (𝐽 ∪ {𝐽}))
6		imaundi 5056	. . . . . 6 ⊢ (𝐹 “ (𝐽 ∪ {𝐽})) = ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽}))
7	5, 6	eqtri 2210	. . . . 5 ⊢ (𝐹 “ suc 𝐽) = ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽}))
8	7	eleq2i 2256	. . . 4 ⊢ (𝑋 ∈ (𝐹 “ suc 𝐽) ↔ 𝑋 ∈ ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽})))
9	3, 8	sylibr 134	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐹 “ 𝐽)) → 𝑋 ∈ (𝐹 “ suc 𝐽))
10	9	orcd 734	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ (𝐹 “ 𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
11		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ 𝑋 = (𝐹‘𝐽)) → 𝑋 = (𝐹‘𝐽))
12		fidcenumlemrks.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
13		elsng 3622	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐴 → (𝑋 ∈ {(𝐹‘𝐽)} ↔ 𝑋 = (𝐹‘𝐽)))
14	12, 13	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ {(𝐹‘𝐽)} ↔ 𝑋 = (𝐹‘𝐽)))
15		fidcenumlemr.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑁–onto→𝐴)
16		fofn 5455	. . . . . . . . . . . 12 ⊢ (𝐹:𝑁–onto→𝐴 → 𝐹 Fn 𝑁)
17	15, 16	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 Fn 𝑁)
18		fidcenumlemrks.jn	. . . . . . . . . . . 12 ⊢ (𝜑 → suc 𝐽 ⊆ 𝑁)
19		fidcenumlemrks.j	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐽 ∈ ω)
20		sucidg 4431	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ ω → 𝐽 ∈ suc 𝐽)
21	19, 20	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ suc 𝐽)
22	18, 21	sseldd 3171	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ 𝑁)
23		fnsnfv 5591	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑁 ∧ 𝐽 ∈ 𝑁) → {(𝐹‘𝐽)} = (𝐹 “ {𝐽}))
24	17, 22, 23	syl2anc 411	. . . . . . . . . 10 ⊢ (𝜑 → {(𝐹‘𝐽)} = (𝐹 “ {𝐽}))
25	24	eleq2d 2259	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ {(𝐹‘𝐽)} ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
26	14, 25	bitr3d 190	. . . . . . . 8 ⊢ (𝜑 → (𝑋 = (𝐹‘𝐽) ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
27	26	ad2antrr 488	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ 𝑋 = (𝐹‘𝐽)) → (𝑋 = (𝐹‘𝐽) ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
28	11, 27	mpbid 147	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ 𝑋 = (𝐹‘𝐽)) → 𝑋 ∈ (𝐹 “ {𝐽}))
29		elun2 3318	. . . . . 6 ⊢ (𝑋 ∈ (𝐹 “ {𝐽}) → 𝑋 ∈ ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽})))
30	28, 29	syl 14	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ 𝑋 = (𝐹‘𝐽)) → 𝑋 ∈ ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽})))
31	30, 8	sylibr 134	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ 𝑋 = (𝐹‘𝐽)) → 𝑋 ∈ (𝐹 “ suc 𝐽))
32	31	orcd 734	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ 𝑋 = (𝐹‘𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
33		simplr 528	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ 𝑋 ∈ (𝐹 “ 𝐽))
34		simpr 110	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ 𝑋 = (𝐹‘𝐽))
35	26	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → (𝑋 = (𝐹‘𝐽) ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
36	34, 35	mtbid 673	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ 𝑋 ∈ (𝐹 “ {𝐽}))
37		ioran 753	. . . . . . 7 ⊢ (¬ (𝑋 ∈ (𝐹 “ 𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})) ↔ (¬ 𝑋 ∈ (𝐹 “ 𝐽) ∧ ¬ 𝑋 ∈ (𝐹 “ {𝐽})))
38	33, 36, 37	sylanbrc 417	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ (𝑋 ∈ (𝐹 “ 𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})))
39		elun 3291	. . . . . 6 ⊢ (𝑋 ∈ ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽})) ↔ (𝑋 ∈ (𝐹 “ 𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})))
40	38, 39	sylnibr 678	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ 𝑋 ∈ ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽})))
41	40, 8	sylnibr 678	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ 𝑋 ∈ (𝐹 “ suc 𝐽))
42	41	olcd 735	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
43		fof 5453	. . . . . . . 8 ⊢ (𝐹:𝑁–onto→𝐴 → 𝐹:𝑁⟶𝐴)
44	15, 43	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑁⟶𝐴)
45	44, 22	ffvelcdmd 5668	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐽) ∈ 𝐴)
46		fidcenumlemr.dc	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
47		eqeq1 2196	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
48	47	dcbid 839	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑋 = 𝑦))
49		eqeq2 2199	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝐽) → (𝑋 = 𝑦 ↔ 𝑋 = (𝐹‘𝐽)))
50	49	dcbid 839	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐽) → (DECID 𝑋 = 𝑦 ↔ DECID 𝑋 = (𝐹‘𝐽)))
51	48, 50	rspc2va 2870	. . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ (𝐹‘𝐽) ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) → DECID 𝑋 = (𝐹‘𝐽))
52	12, 45, 46, 51	syl21anc 1248	. . . . 5 ⊢ (𝜑 → DECID 𝑋 = (𝐹‘𝐽))
53		exmiddc 837	. . . . 5 ⊢ (DECID 𝑋 = (𝐹‘𝐽) → (𝑋 = (𝐹‘𝐽) ∨ ¬ 𝑋 = (𝐹‘𝐽)))
54	52, 53	syl 14	. . . 4 ⊢ (𝜑 → (𝑋 = (𝐹‘𝐽) ∨ ¬ 𝑋 = (𝐹‘𝐽)))
55	54	adantr 276	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) → (𝑋 = (𝐹‘𝐽) ∨ ¬ 𝑋 = (𝐹‘𝐽)))
56	32, 42, 55	mpjaodan 799	. 2 ⊢ ((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
57		fidcenumlemrks.h	. 2 ⊢ (𝜑 → (𝑋 ∈ (𝐹 “ 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐽)))
58	10, 56, 57	mpjaodan 799	1 ⊢ (𝜑 → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))

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  –onto→wfo 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