Theorem fidcenumlemrk 7020

Description: Lemma for fidcenum 7022. (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 3206	. . . . 5 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝑁 ↔ ∅ ⊆ 𝑁))
5	4	anbi2d 464	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ ∅ ⊆ 𝑁)))
6		imaeq2 5005	. . . . . 6 ⊢ (𝑤 = ∅ → (𝐹 “ 𝑤) = (𝐹 “ ∅))
7	6	eleq2d 2266	. . . . 5 ⊢ (𝑤 = ∅ → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ ∅)))
8	7	notbid 668	. . . . 5 ⊢ (𝑤 = ∅ → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ ∅)))
9	7, 8	orbi12d 794	. . . 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 3206	. . . . 5 ⊢ (𝑤 = 𝑗 → (𝑤 ⊆ 𝑁 ↔ 𝑗 ⊆ 𝑁))
12	11	anbi2d 464	. . . 4 ⊢ (𝑤 = 𝑗 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝑗 ⊆ 𝑁)))
13		imaeq2 5005	. . . . . 6 ⊢ (𝑤 = 𝑗 → (𝐹 “ 𝑤) = (𝐹 “ 𝑗))
14	13	eleq2d 2266	. . . . 5 ⊢ (𝑤 = 𝑗 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ 𝑗)))
15	14	notbid 668	. . . . 5 ⊢ (𝑤 = 𝑗 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))
16	14, 15	orbi12d 794	. . . 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 3206	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (𝑤 ⊆ 𝑁 ↔ suc 𝑗 ⊆ 𝑁))
19	18	anbi2d 464	. . . 4 ⊢ (𝑤 = suc 𝑗 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)))
20		imaeq2 5005	. . . . . 6 ⊢ (𝑤 = suc 𝑗 → (𝐹 “ 𝑤) = (𝐹 “ suc 𝑗))
21	20	eleq2d 2266	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ suc 𝑗)))
22	21	notbid 668	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ suc 𝑗)))
23	21, 22	orbi12d 794	. . . 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 3206	. . . . 5 ⊢ (𝑤 = 𝐾 → (𝑤 ⊆ 𝑁 ↔ 𝐾 ⊆ 𝑁))
26	25	anbi2d 464	. . . 4 ⊢ (𝑤 = 𝐾 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝐾 ⊆ 𝑁)))
27		imaeq2 5005	. . . . . 6 ⊢ (𝑤 = 𝐾 → (𝐹 “ 𝑤) = (𝐹 “ 𝐾))
28	27	eleq2d 2266	. . . . 5 ⊢ (𝑤 = 𝐾 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ 𝐾)))
29	28	notbid 668	. . . . 5 ⊢ (𝑤 = 𝐾 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ 𝐾)))
30	28, 29	orbi12d 794	. . . 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 3454	. . . . . 6 ⊢ ¬ 𝑋 ∈ ∅
33		ima0 5028	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
34	33	eleq2i 2263	. . . . . 6 ⊢ (𝑋 ∈ (𝐹 “ ∅) ↔ 𝑋 ∈ ∅)
35	32, 34	mtbir 672	. . . . 5 ⊢ ¬ 𝑋 ∈ (𝐹 “ ∅)
36	35	a1i 9	. . . 4 ⊢ ((𝜑 ∧ ∅ ⊆ 𝑁) → ¬ 𝑋 ∈ (𝐹 “ ∅))
37	36	olcd 735	. . 3 ⊢ ((𝜑 ∧ ∅ ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ ∅) ∨ ¬ 𝑋 ∈ (𝐹 “ ∅)))
38		fidcenumlemr.dc	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
39	38	ad2antrl 490	. . . . 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 490	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝐹:𝑁–onto→𝐴)
42		simpll 527	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝑗 ∈ ω)
43		simprr 531	. . . . 5 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → suc 𝑗 ⊆ 𝑁)
44		simprl 529	. . . . . 6 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝜑)
45		sssucid 4450	. . . . . . 7 ⊢ 𝑗 ⊆ suc 𝑗
46	45, 43	sstrid 3194	. . . . . 6 ⊢ (((𝑗 ∈ ω ∧ ((𝜑 ∧ 𝑗 ⊆ 𝑁) → (𝑋 ∈ (𝐹 “ 𝑗) ∨ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))) ∧ (𝜑 ∧ suc 𝑗 ⊆ 𝑁)) → 𝑗 ⊆ 𝑁)
47		simplr 528	. . . . . 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 433	. . . . 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 490	. . . . 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 7019	. . . 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 4636	. 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 104   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2167  ∀wral 2475   ⊆ wss 3157  ∅c0 3450  suc csuc 4400  ωcom 4626   “ cima 4666  –onto→wfo 5256

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fo 5264  df-fv 5266

This theorem is referenced by:  fidcenumlemr  7021  ennnfonelemdc  12616