Theorem fidcenumlemrk 7271

Description: Lemma for fidcenum 7273. (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)
fidcenumlemrk.k	⊢ (𝜑 → 𝐾 ∈ ω)
fidcenumlemrk.kn	⊢ (𝜑 → 𝐾 ⊆ 𝑁)
fidcenumlemrk.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
fidcenumlemrk	⊢ (𝜑 → (𝑋 ∈ (𝐹 “ 𝐾) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐾)))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝑋,𝑦   𝑥,𝐴,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐾(𝑥, 𝑦)   𝑁(𝑥, 𝑦)

Proof of Theorem fidcenumlemrk

Dummy variables 𝑗 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fidcenumlemrk.k	. 2 ⊢ (𝜑 → 𝐾 ∈ ω)
2		fidcenumlemrk.kn	. . 3 ⊢ (𝜑 → 𝐾 ⊆ 𝑁)
3	2	ancli 323	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝐾 ⊆ 𝑁))
4		sseq1 3271	. . . . 5 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝑁 ↔ ∅ ⊆ 𝑁))
5	4	anbi2d 468	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ ∅ ⊆ 𝑁)))
6		imaeq2 5122	. . . . . 6 ⊢ (𝑤 = ∅ → (𝐹 “ 𝑤) = (𝐹 “ ∅))
7	6	eleq2d 2308	. . . . 5 ⊢ (𝑤 = ∅ → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ ∅)))
8	7	notbid 677	. . . . 5 ⊢ (𝑤 = ∅ → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ ∅)))
9	7, 8	orbi12d 805	. . . 4 ⊢ (𝑤 = ∅ → ((𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤)) ↔ (𝑋 ∈ (𝐹 “ ∅) ∨ ¬ 𝑋 ∈ (𝐹 “ ∅))))
10	5, 9	imbi12d 234	. . 3 ⊢ (𝑤 = ∅ → (((𝜑 ∧ 𝑤 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤))) ↔ ((𝜑 ∧ ∅ ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ ∅) ∨ ¬ 𝑋 ∈ (𝐹 “ ∅)))))
11		sseq1 3271	. . . . 5 ⊢ (𝑤 = 𝑗 → (𝑤 ⊆ 𝑁 ↔ 𝑗 ⊆ 𝑁))
12	11	anbi2d 468	. . . 4 ⊢ (𝑤 = 𝑗 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝑗 ⊆ 𝑁)))
13		imaeq2 5122	. . . . . 6 ⊢ (𝑤 = 𝑗 → (𝐹 “ 𝑤) = (𝐹 “ 𝑗))
14	13	eleq2d 2308	. . . . 5 ⊢ (𝑤 = 𝑗 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ 𝑗)))
15	14	notbid 677	. . . . 5 ⊢ (𝑤 = 𝑗 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))
16	14, 15	orbi12d 805	. . . 4 ⊢ (𝑤 = 𝑗 → ((𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤)) ↔ (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗))))
17	12, 16	imbi12d 234	. . 3 ⊢ (𝑤 = 𝑗 → (((𝜑 ∧ 𝑤 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤))) ↔ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))))
18		sseq1 3271	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (𝑤 ⊆ 𝑁 ↔ suc 𝑗 ⊆ 𝑁))
19	18	anbi2d 468	. . . 4 ⊢ (𝑤 = suc 𝑗 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)))
20		imaeq2 5122	. . . . . 6 ⊢ (𝑤 = suc 𝑗 → (𝐹 “ 𝑤) = (𝐹 “ suc 𝑗))
21	20	eleq2d 2308	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ suc 𝑗)))
22	21	notbid 677	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ suc 𝑗)))
23	21, 22	orbi12d 805	. . . 4 ⊢ (𝑤 = suc 𝑗 → ((𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤)) ↔ (𝑋 ∈ (𝐹 “ suc 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝑗))))
24	19, 23	imbi12d 234	. . 3 ⊢ (𝑤 = suc 𝑗 → (((𝜑 ∧ 𝑤 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤))) ↔ ((𝜑 ∧ suc 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ suc 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝑗)))))
25		sseq1 3271	. . . . 5 ⊢ (𝑤 = 𝐾 → (𝑤 ⊆ 𝑁 ↔ 𝐾 ⊆ 𝑁))
26	25	anbi2d 468	. . . 4 ⊢ (𝑤 = 𝐾 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝐾 ⊆ 𝑁)))
27		imaeq2 5122	. . . . . 6 ⊢ (𝑤 = 𝐾 → (𝐹 “ 𝑤) = (𝐹 “ 𝐾))
28	27	eleq2d 2308	. . . . 5 ⊢ (𝑤 = 𝐾 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ 𝐾)))
29	28	notbid 677	. . . . 5 ⊢ (𝑤 = 𝐾 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ 𝐾)))
30	28, 29	orbi12d 805	. . . 4 ⊢ (𝑤 = 𝐾 → ((𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤)) ↔ (𝑋 ∈ (𝐹 “ 𝐾) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐾))))
31	26, 30	imbi12d 234	. . 3 ⊢ (𝑤 = 𝐾 → (((𝜑 ∧ 𝑤 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤))) ↔ ((𝜑 ∧ 𝐾 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝐾) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐾)))))
32		noel 3525	. . . . . 6 ⊢ ¬ 𝑋 ∈ ∅
33		ima0 5146	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
34	33	eleq2i 2305	. . . . . 6 ⊢ (𝑋 ∈ (𝐹 “ ∅) ↔ 𝑋 ∈ ∅)
35	32, 34	mtbir 682	. . . . 5 ⊢ ¬ 𝑋 ∈ (𝐹 “ ∅)
36	35	a1i 9	. . . 4 ⊢ ((𝜑 ∧ ∅ ⊆ 𝑁) → ¬ 𝑋 ∈ (𝐹 “ ∅))
37	36	olcd 746	. . 3 ⊢ ((𝜑 ∧ ∅ ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ ∅) ∨ ¬ 𝑋 ∈ (𝐹 “ ∅)))
38		fidcenumlemr.dc	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
39	38	ad2antrl 494	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
40		fidcenumlemr.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝑁–onto→𝐴)
41	40	ad2antrl 494	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝐹:𝑁–onto→𝐴)
42		simpll 531	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝑗 ∈ ω)
43		simprr 537	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → suc 𝑗 ⊆ 𝑁)
44		simprl 535	. . . . . 6 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝜑)
45		sssucid 4560	. . . . . . 7 ⊢ 𝑗 ⊆ suc 𝑗
46	45, 43	sstrid 3259	. . . . . 6 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝑗 ⊆ 𝑁)
47		simplr 533	. . . . . 6 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗))))
48	44, 46, 47	mp2and 437	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))
49		fidcenumlemrk.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
50	49	ad2antrl 494	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝑋 ∈ 𝐴)
51	39, 41, 42, 43, 48, 50	fidcenumlemrks 7270	. . . 4 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → (𝑋 ∈ (𝐹 “ suc 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝑗)))
52	51	exp31 364	. . 3 ⊢ (𝑗 ∈ ω → (((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗))) → ((𝜑 ∧ suc 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ suc 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝑗)))))
53	10, 17, 24, 31, 37, 52	finds 4747	. 2 ⊢ (𝐾 ∈ ω → ((𝜑 ∧ 𝐾 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝐾) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐾))))
54	1, 3, 53	sylc 62	1 ⊢ (𝜑 → (𝑋 ∈ (𝐹 “ 𝐾) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐾)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  ∅c0 3520  suc csuc 4510  ωcom 4737   “ cima 4777  –onto→wfo 5375

This proof depends on 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 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3an 1011  df-tru 1405  df-fal 1408  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-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fo 5383  df-fv 5385

This theorem is used by:  fidcenumlemr  7272  ennnfonelemdc  13292