Theorem fidcenumlemrks 7264

Description: Lemma for fidcenum 7267. Induction step for fidcenumlemrk 7265. (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 3396	. . . . 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 4514	. . . . . . 7 ⊢ suc 𝐽 = (𝐽 ∪ {𝐽})
5	4	imaeq2i 5122	. . . . . 6 ⊢ (𝐹 “ suc 𝐽) = (𝐹 “ (𝐽 ∪ {𝐽}))
6		imaundi 5198	. . . . . 6 ⊢ (𝐹 “ (𝐽 ∪ {𝐽})) = ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽}))
7	5, 6	eqtri 2259	. . . . 5 ⊢ (𝐹 “ suc 𝐽) = ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽}))
8	7	eleq2i 2305	. . . 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 745	. 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 3723	. . . . . . . . . 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 5615	. . . . . . . . . . . 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 4559	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ ω → 𝐽 ∈ suc 𝐽)
21	19, 20	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ suc 𝐽)
22	18, 21	sseldd 3249	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ 𝑁)
23		fnsnfv 5759	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝑁 ∧ 𝐽 ∈ 𝑁) → {(𝐹‘𝐽)} = (𝐹 “ {𝐽}))
24	17, 22, 23	syl2anc 415	. . . . . . . . . 10 ⊢ (𝜑 → {(𝐹‘𝐽)} = (𝐹 “ {𝐽}))
25	24	eleq2d 2308	. . . . . . . . 9 ⊢ (𝜑 → (𝑋 ∈ {(𝐹‘𝐽)} ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
26	14, 25	bitr3d 190	. . . . . . . 8 ⊢ (𝜑 → (𝑋 = (𝐹‘𝐽) ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
27	26	ad2antrr 492	. . . . . . 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 3397	. . . . . 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 745	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ 𝑋 = (𝐹‘𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
33		simplr 533	. . . . . . 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 492	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → (𝑋 = (𝐹‘𝐽) ↔ 𝑋 ∈ (𝐹 “ {𝐽})))
36	34, 35	mtbid 683	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ 𝑋 ∈ (𝐹 “ {𝐽}))
37		ioran 764	. . . . . . 7 ⊢ (¬ (𝑋 ∈ (𝐹 “ 𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})) ↔ (¬ 𝑋 ∈ (𝐹 “ 𝐽) ∧ ¬ 𝑋 ∈ (𝐹 “ {𝐽})))
38	33, 36, 37	sylanbrc 421	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ (𝑋 ∈ (𝐹 “ 𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})))
39		elun 3370	. . . . . 6 ⊢ (𝑋 ∈ ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽})) ↔ (𝑋 ∈ (𝐹 “ 𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})))
40	38, 39	sylnibr 688	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ 𝑋 ∈ ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽})))
41	40, 8	sylnibr 688	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ 𝑋 ∈ (𝐹 “ suc 𝐽))
42	41	olcd 746	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
43		fof 5613	. . . . . . . 8 ⊢ (𝐹:𝑁–onto→𝐴 → 𝐹:𝑁⟶𝐴)
44	15, 43	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑁⟶𝐴)
45	44, 22	ffvelcdmd 5838	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐽) ∈ 𝐴)
46		fidcenumlemr.dc	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
47		eqeq1 2245	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
48	47	dcbid 850	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑋 = 𝑦))
49		eqeq2 2248	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝐽) → (𝑋 = 𝑦 ↔ 𝑋 = (𝐹‘𝐽)))
50	49	dcbid 850	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐽) → (DECID 𝑋 = 𝑦 ↔ DECID 𝑋 = (𝐹‘𝐽)))
51	48, 50	rspc2va 2944	. . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ (𝐹‘𝐽) ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) → DECID 𝑋 = (𝐹‘𝐽))
52	12, 45, 46, 51	syl21anc 1277	. . . . 5 ⊢ (𝜑 → DECID 𝑋 = (𝐹‘𝐽))
53		exmiddc 848	. . . . 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 810	. 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 810	1 ⊢ (𝜑 → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ∪ cun 3218   ⊆ wss 3220  {csn 3708  suc csuc 4508  ωcom 4735   “ cima 4775   Fn wfn 5370  ⟶wf 5371  –onto→wfo 5373  ‘cfv 5375

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-suc 4514  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fo 5381  df-fv 5383

This theorem is referenced by:  fidcenumlemrk  7265