Theorem fidcenumlemrks 7152

Description: Lemma for fidcenum 7155. Induction step for fidcenumlemrk 7153. (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 3374	. . . . 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 4468	. . . . . . 7 ⊢ suc 𝐽 = (𝐽 ∪ {𝐽})
5	4	imaeq2i 5074	. . . . . 6 ⊢ (𝐹 “ suc 𝐽) = (𝐹 “ (𝐽 ∪ {𝐽}))
6		imaundi 5149	. . . . . 6 ⊢ (𝐹 “ (𝐽 ∪ {𝐽})) = ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽}))
7	5, 6	eqtri 2252	. . . . 5 ⊢ (𝐹 “ suc 𝐽) = ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽}))
8	7	eleq2i 2298	. . . 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 740	. 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 3684	. . . . . . . . . 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 5561	. . . . . . . . . . . 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 4513	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ ω → 𝐽 ∈ suc 𝐽)
21	19, 20	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽 ∈ suc 𝐽)
22	18, 21	sseldd 3228	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐽 ∈ 𝑁)
23		fnsnfv 5705	. . . . . . . . . . 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 2301	. . . . . . . . 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 3375	. . . . . 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 740	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ 𝑋 = (𝐹‘𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
33		simplr 529	. . . . . . 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 678	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ 𝑋 ∈ (𝐹 “ {𝐽}))
37		ioran 759	. . . . . . 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 3348	. . . . . 6 ⊢ (𝑋 ∈ ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽})) ↔ (𝑋 ∈ (𝐹 “ 𝐽) ∨ 𝑋 ∈ (𝐹 “ {𝐽})))
40	38, 39	sylnibr 683	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ 𝑋 ∈ ((𝐹 “ 𝐽) ∪ (𝐹 “ {𝐽})))
41	40, 8	sylnibr 683	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → ¬ 𝑋 ∈ (𝐹 “ suc 𝐽))
42	41	olcd 741	. . 3 ⊢ (((𝜑 ∧ ¬ 𝑋 ∈ (𝐹 “ 𝐽)) ∧ ¬ 𝑋 = (𝐹‘𝐽)) → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))
43		fof 5559	. . . . . . . 8 ⊢ (𝐹:𝑁–onto→𝐴 → 𝐹:𝑁⟶𝐴)
44	15, 43	syl 14	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑁⟶𝐴)
45	44, 22	ffvelcdmd 5783	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝐽) ∈ 𝐴)
46		fidcenumlemr.dc	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
47		eqeq1 2238	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
48	47	dcbid 845	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑋 = 𝑦))
49		eqeq2 2241	. . . . . . . 8 ⊢ (𝑦 = (𝐹‘𝐽) → (𝑋 = 𝑦 ↔ 𝑋 = (𝐹‘𝐽)))
50	49	dcbid 845	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝐽) → (DECID 𝑋 = 𝑦 ↔ DECID 𝑋 = (𝐹‘𝐽)))
51	48, 50	rspc2va 2924	. . . . . 6 ⊢ (((𝑋 ∈ 𝐴 ∧ (𝐹‘𝐽) ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) → DECID 𝑋 = (𝐹‘𝐽))
52	12, 45, 46, 51	syl21anc 1272	. . . . 5 ⊢ (𝜑 → DECID 𝑋 = (𝐹‘𝐽))
53		exmiddc 843	. . . . 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 805	. 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 805	1 ⊢ (𝜑 → (𝑋 ∈ (𝐹 “ suc 𝐽) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝐽)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715  DECID wdc 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510   ∪ cun 3198   ⊆ wss 3200  {csn 3669  suc csuc 4462  ωcom 4688   “ cima 4728   Fn wfn 5321  ⟶wf 5322  –onto→wfo 5324  ‘cfv 5326

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334

This theorem is referenced by:  fidcenumlemrk  7153