Theorem fidcenumlemrk 6931

Description: Lemma for fidcenum 6933. (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 321	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝐾 ⊆ 𝑁))
4		sseq1 3170	. . . . 5 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝑁 ↔ ∅ ⊆ 𝑁))
5	4	anbi2d 461	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ ∅ ⊆ 𝑁)))
6		imaeq2 4949	. . . . . 6 ⊢ (𝑤 = ∅ → (𝐹 “ 𝑤) = (𝐹 “ ∅))
7	6	eleq2d 2240	. . . . 5 ⊢ (𝑤 = ∅ → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ ∅)))
8	7	notbid 662	. . . . 5 ⊢ (𝑤 = ∅ → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ ∅)))
9	7, 8	orbi12d 788	. . . 4 ⊢ (𝑤 = ∅ → ((𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤)) ↔ (𝑋 ∈ (𝐹 “ ∅) ∨ ¬ 𝑋 ∈ (𝐹 “ ∅))))
10	5, 9	imbi12d 233	. . 3 ⊢ (𝑤 = ∅ → (((𝜑 ∧ 𝑤 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤))) ↔ ((𝜑 ∧ ∅ ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ ∅) ∨ ¬ 𝑋 ∈ (𝐹 “ ∅)))))
11		sseq1 3170	. . . . 5 ⊢ (𝑤 = 𝑗 → (𝑤 ⊆ 𝑁 ↔ 𝑗 ⊆ 𝑁))
12	11	anbi2d 461	. . . 4 ⊢ (𝑤 = 𝑗 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝑗 ⊆ 𝑁)))
13		imaeq2 4949	. . . . . 6 ⊢ (𝑤 = 𝑗 → (𝐹 “ 𝑤) = (𝐹 “ 𝑗))
14	13	eleq2d 2240	. . . . 5 ⊢ (𝑤 = 𝑗 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ 𝑗)))
15	14	notbid 662	. . . . 5 ⊢ (𝑤 = 𝑗 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))
16	14, 15	orbi12d 788	. . . 4 ⊢ (𝑤 = 𝑗 → ((𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤)) ↔ (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗))))
17	12, 16	imbi12d 233	. . 3 ⊢ (𝑤 = 𝑗 → (((𝜑 ∧ 𝑤 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤))) ↔ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))))
18		sseq1 3170	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (𝑤 ⊆ 𝑁 ↔ suc 𝑗 ⊆ 𝑁))
19	18	anbi2d 461	. . . 4 ⊢ (𝑤 = suc 𝑗 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)))
20		imaeq2 4949	. . . . . 6 ⊢ (𝑤 = suc 𝑗 → (𝐹 “ 𝑤) = (𝐹 “ suc 𝑗))
21	20	eleq2d 2240	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ suc 𝑗)))
22	21	notbid 662	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ suc 𝑗)))
23	21, 22	orbi12d 788	. . . 4 ⊢ (𝑤 = suc 𝑗 → ((𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤)) ↔ (𝑋 ∈ (𝐹 “ suc 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ suc 𝑗))))
24	19, 23	imbi12d 233	. . 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 3170	. . . . 5 ⊢ (𝑤 = 𝐾 → (𝑤 ⊆ 𝑁 ↔ 𝐾 ⊆ 𝑁))
26	25	anbi2d 461	. . . 4 ⊢ (𝑤 = 𝐾 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝐾 ⊆ 𝑁)))
27		imaeq2 4949	. . . . . 6 ⊢ (𝑤 = 𝐾 → (𝐹 “ 𝑤) = (𝐹 “ 𝐾))
28	27	eleq2d 2240	. . . . 5 ⊢ (𝑤 = 𝐾 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ 𝐾)))
29	28	notbid 662	. . . . 5 ⊢ (𝑤 = 𝐾 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ 𝐾)))
30	28, 29	orbi12d 788	. . . 4 ⊢ (𝑤 = 𝐾 → ((𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤)) ↔ (𝑋 ∈ (𝐹 “ 𝐾) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐾))))
31	26, 30	imbi12d 233	. . 3 ⊢ (𝑤 = 𝐾 → (((𝜑 ∧ 𝑤 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑤) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑤))) ↔ ((𝜑 ∧ 𝐾 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝐾) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝐾)))))
32		noel 3418	. . . . . 6 ⊢ ¬ 𝑋 ∈ ∅
33		ima0 4970	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
34	33	eleq2i 2237	. . . . . 6 ⊢ (𝑋 ∈ (𝐹 “ ∅) ↔ 𝑋 ∈ ∅)
35	32, 34	mtbir 666	. . . . 5 ⊢ ¬ 𝑋 ∈ (𝐹 “ ∅)
36	35	a1i 9	. . . 4 ⊢ ((𝜑 ∧ ∅ ⊆ 𝑁) → ¬ 𝑋 ∈ (𝐹 “ ∅))
37	36	olcd 729	. . 3 ⊢ ((𝜑 ∧ ∅ ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ ∅) ∨ ¬ 𝑋 ∈ (𝐹 “ ∅)))
38		fidcenumlemr.dc	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
39	38	ad2antrl 487	. . . . 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 487	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝐹:𝑁–onto→𝐴)
42		simpll 524	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝑗 ∈ ω)
43		simprr 527	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → suc 𝑗 ⊆ 𝑁)
44		simprl 526	. . . . . 6 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝜑)
45		sssucid 4400	. . . . . . 7 ⊢ 𝑗 ⊆ suc 𝑗
46	45, 43	sstrid 3158	. . . . . 6 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝑗 ⊆ 𝑁)
47		simplr 525	. . . . . 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 431	. . . . 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 487	. . . . 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 6930	. . . 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 362	. . 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 4584	. 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)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∨ wo 703  DECID wdc 829   = wceq 1348   ∈ wcel 2141  ∀wral 2448   ⊆ wss 3121  ∅c0 3414  suc csuc 4350  ωcom 4574   “ cima 4614  –onto→wfo 5196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fo 5204  df-fv 5206

This theorem is referenced by:  fidcenumlemr  6932  ennnfonelemdc  12354