Theorem fidcenumlemrk 6947

Description: Lemma for fidcenum 6949. (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 3178	. . . . 5 ⊢ (𝑤 = ∅ → (𝑤 ⊆ 𝑁 ↔ ∅ ⊆ 𝑁))
5	4	anbi2d 464	. . . 4 ⊢ (𝑤 = ∅ → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ ∅ ⊆ 𝑁)))
6		imaeq2 4962	. . . . . 6 ⊢ (𝑤 = ∅ → (𝐹 “ 𝑤) = (𝐹 “ ∅))
7	6	eleq2d 2247	. . . . 5 ⊢ (𝑤 = ∅ → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ ∅)))
8	7	notbid 667	. . . . 5 ⊢ (𝑤 = ∅ → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ ∅)))
9	7, 8	orbi12d 793	. . . 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 3178	. . . . 5 ⊢ (𝑤 = 𝑗 → (𝑤 ⊆ 𝑁 ↔ 𝑗 ⊆ 𝑁))
12	11	anbi2d 464	. . . 4 ⊢ (𝑤 = 𝑗 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝑗 ⊆ 𝑁)))
13		imaeq2 4962	. . . . . 6 ⊢ (𝑤 = 𝑗 → (𝐹 “ 𝑤) = (𝐹 “ 𝑗))
14	13	eleq2d 2247	. . . . 5 ⊢ (𝑤 = 𝑗 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ 𝑗)))
15	14	notbid 667	. . . . 5 ⊢ (𝑤 = 𝑗 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ 𝑗)))
16	14, 15	orbi12d 793	. . . 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 3178	. . . . 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 4962	. . . . . 6 ⊢ (𝑤 = suc 𝑗 → (𝐹 “ 𝑤) = (𝐹 “ suc 𝑗))
21	20	eleq2d 2247	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ suc 𝑗)))
22	21	notbid 667	. . . . 5 ⊢ (𝑤 = suc 𝑗 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ suc 𝑗)))
23	21, 22	orbi12d 793	. . . 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 3178	. . . . 5 ⊢ (𝑤 = 𝐾 → (𝑤 ⊆ 𝑁 ↔ 𝐾 ⊆ 𝑁))
26	25	anbi2d 464	. . . 4 ⊢ (𝑤 = 𝐾 → ((𝜑 ∧ 𝑤 ⊆ 𝑁) ↔ (𝜑 ∧ 𝐾 ⊆ 𝑁)))
27		imaeq2 4962	. . . . . 6 ⊢ (𝑤 = 𝐾 → (𝐹 “ 𝑤) = (𝐹 “ 𝐾))
28	27	eleq2d 2247	. . . . 5 ⊢ (𝑤 = 𝐾 → (𝑋 ∈ (𝐹 “ 𝑤) ↔ 𝑋 ∈ (𝐹 “ 𝐾)))
29	28	notbid 667	. . . . 5 ⊢ (𝑤 = 𝐾 → (¬ 𝑋 ∈ (𝐹 “ 𝑤) ↔ ¬ 𝑋 ∈ (𝐹 “ 𝐾)))
30	28, 29	orbi12d 793	. . . 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 3426	. . . . . 6 ⊢ ¬ 𝑋 ∈ ∅
33		ima0 4983	. . . . . . 7 ⊢ (𝐹 “ ∅) = ∅
34	33	eleq2i 2244	. . . . . 6 ⊢ (𝑋 ∈ (𝐹 “ ∅) ↔ 𝑋 ∈ ∅)
35	32, 34	mtbir 671	. . . . 5 ⊢ ¬ 𝑋 ∈ (𝐹 “ ∅)
36	35	a1i 9	. . . 4 ⊢ ((𝜑 ∧ ∅ ⊆ 𝑁) → ¬ 𝑋 ∈ (𝐹 “ ∅))
37	36	olcd 734	. . 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 4412	. . . . . . 7 ⊢ 𝑗 ⊆ suc 𝑗
46	45, 43	sstrid 3166	. . . . . 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 6946	. . . 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 4596	. 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 708  DECID wdc 834   = wceq 1353   ∈ wcel 2148  ∀wral 2455   ⊆ wss 3129  ∅c0 3422  suc csuc 4362  ωcom 4586   “ cima 4626  –onto→wfo 5210

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4290  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-fo 5218  df-fv 5220

This theorem is referenced by:  fidcenumlemr  6948  ennnfonelemdc  12383