Theorem pwfseqlem3 10697

Description: Lemma for pwfseq 10701. Using the construction 𝐷 from pwfseqlem1 10695, produce a function 𝐹 that maps any well-ordered infinite set to an element outside the set. (Contributed by Mario Carneiro, 31-May-2015.)

Hypotheses

Ref	Expression
pwfseqlem4.g	⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
pwfseqlem4.x	⊢ (𝜑 → 𝑋 ⊆ 𝐴)
pwfseqlem4.h	⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)
pwfseqlem4.ps	⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
pwfseqlem4.k	⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
pwfseqlem4.d	⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
pwfseqlem4.f	⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))

Assertion

Ref	Expression
pwfseqlem3	⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥))

Distinct variable groups:   𝑛,𝑟,𝑤,𝑥,𝑧   𝐷,𝑛,𝑧   𝑤,𝐺   𝑤,𝐾   𝐻,𝑟,𝑥,𝑧   𝜑,𝑛,𝑟,𝑥,𝑧   𝜓,𝑛,𝑧   𝐴,𝑛,𝑟,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑥,𝑤,𝑟)   𝐴(𝑤)   𝐷(𝑥,𝑤,𝑟)   𝐹(𝑥,𝑧,𝑤,𝑛,𝑟)   𝐺(𝑥,𝑧,𝑛,𝑟)   𝐻(𝑤,𝑛)   𝐾(𝑥,𝑧,𝑛,𝑟)   𝑋(𝑥,𝑧,𝑤,𝑛,𝑟)

Proof of Theorem pwfseqlem3

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3481	. . . 4 ⊢ 𝑥 ∈ V
2		vex 3481	. . . 4 ⊢ 𝑟 ∈ V
3		fvex 6919	. . . . 5 ⊢ (𝐻‘(card‘𝑥)) ∈ V
4		fvex 6919	. . . . 5 ⊢ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ V
5	3, 4	ifex 4580	. . . 4 ⊢ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})) ∈ V
6		pwfseqlem4.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
7	6	ovmpt4g 7579	. . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑟 ∈ V ∧ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})) ∈ V) → (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})))
8	1, 2, 5, 7	mp3an 1460	. . 3 ⊢ (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))
9		pwfseqlem4.ps	. . . . . . . 8 ⊢ (𝜓 ↔ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
10	9	simprbi 496	. . . . . . 7 ⊢ (𝜓 → ω ≼ 𝑥)
11	10	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ 𝑥)
12		domnsym 9137	. . . . . 6 ⊢ (ω ≼ 𝑥 → ¬ 𝑥 ≺ ω)
13	11, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑥 ≺ ω)
14		isfinite 9689	. . . . 5 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
15	13, 14	sylnibr 329	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑥 ∈ Fin)
16	15	iffalsed 4541	. . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})) = (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))
17	8, 16	eqtrid 2786	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) = (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))
18		pwfseqlem4.g	. . . . . . 7 ⊢ (𝜑 → 𝐺:𝒫 𝐴–1-1→∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
19		pwfseqlem4.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝐴)
20		pwfseqlem4.h	. . . . . . 7 ⊢ (𝜑 → 𝐻:ω–1-1-onto→𝑋)
21		pwfseqlem4.k	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → 𝐾:∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)–1-1→𝑥)
22		pwfseqlem4.d	. . . . . . 7 ⊢ 𝐷 = (𝐺‘{𝑤 ∈ 𝑥 ∣ ((◡𝐾‘𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (◡𝐺‘(◡𝐾‘𝑤)))})
23	18, 19, 20, 9, 21, 22	pwfseqlem1 10695	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ (∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∖ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)))
24		eldif 3972	. . . . . 6 ⊢ (𝐷 ∈ (∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∖ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)) ↔ (𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ ¬ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)))
25	23, 24	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ∧ ¬ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛)))
26	25	simpld 494	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛))
27		eliun 4999	. . . 4 ⊢ (𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑m 𝑛) ↔ ∃𝑛 ∈ ω 𝐷 ∈ (𝐴 ↑m 𝑛))
28	26, 27	sylib 218	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑛 ∈ ω 𝐷 ∈ (𝐴 ↑m 𝑛))
29		elmapi 8887	. . . . . 6 ⊢ (𝐷 ∈ (𝐴 ↑m 𝑛) → 𝐷:𝑛⟶𝐴)
30	29	ad2antll 729	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → 𝐷:𝑛⟶𝐴)
31		ssiun2 5051	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → (𝑥 ↑m 𝑛) ⊆ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛))
32	31	ad2antrl 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → (𝑥 ↑m 𝑛) ⊆ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛))
33	25	simprd 495	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛))
34	33	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → ¬ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑m 𝑛))
35	32, 34	ssneldd 3997	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → ¬ 𝐷 ∈ (𝑥 ↑m 𝑛))
36		vex 3481	. . . . . . . . 9 ⊢ 𝑛 ∈ V
37	1, 36	elmap 8909	. . . . . . . 8 ⊢ (𝐷 ∈ (𝑥 ↑m 𝑛) ↔ 𝐷:𝑛⟶𝑥)
38		ffn 6736	. . . . . . . . 9 ⊢ (𝐷:𝑛⟶𝐴 → 𝐷 Fn 𝑛)
39		ffnfv 7138	. . . . . . . . . 10 ⊢ (𝐷:𝑛⟶𝑥 ↔ (𝐷 Fn 𝑛 ∧ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
40	39	baib 535	. . . . . . . . 9 ⊢ (𝐷 Fn 𝑛 → (𝐷:𝑛⟶𝑥 ↔ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
41	30, 38, 40	3syl 18	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → (𝐷:𝑛⟶𝑥 ↔ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
42	37, 41	bitrid 283	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → (𝐷 ∈ (𝑥 ↑m 𝑛) ↔ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
43	35, 42	mtbid 324	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → ¬ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥)
44		nnon 7892	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
45	44	ad2antrl 728	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → 𝑛 ∈ On)
46		ssrab2 4089	. . . . . . . . . 10 ⊢ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ⊆ ω
47		omsson 7890	. . . . . . . . . 10 ⊢ ω ⊆ On
48	46, 47	sstri 4004	. . . . . . . . 9 ⊢ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ⊆ On
49		ordom 7896	. . . . . . . . . . . . 13 ⊢ Ord ω
50		simprl 771	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → 𝑛 ∈ ω)
51		ordelss 6401	. . . . . . . . . . . . 13 ⊢ ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
52	49, 50, 51	sylancr 587	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → 𝑛 ⊆ ω)
53		rexnal 3097	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ 𝑛 ¬ (𝐷‘𝑧) ∈ 𝑥 ↔ ¬ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥)
54	43, 53	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → ∃𝑧 ∈ 𝑛 ¬ (𝐷‘𝑧) ∈ 𝑥)
55		ssrexv 4064	. . . . . . . . . . . 12 ⊢ (𝑛 ⊆ ω → (∃𝑧 ∈ 𝑛 ¬ (𝐷‘𝑧) ∈ 𝑥 → ∃𝑧 ∈ ω ¬ (𝐷‘𝑧) ∈ 𝑥))
56	52, 54, 55	sylc 65	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → ∃𝑧 ∈ ω ¬ (𝐷‘𝑧) ∈ 𝑥)
57		rabn0 4394	. . . . . . . . . . 11 ⊢ ({𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ≠ ∅ ↔ ∃𝑧 ∈ ω ¬ (𝐷‘𝑧) ∈ 𝑥)
58	56, 57	sylibr 234	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ≠ ∅)
59		onint 7809	. . . . . . . . . 10 ⊢ (({𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ⊆ On ∧ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ≠ ∅) → ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})
60	48, 58, 59	sylancr 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})
61	48, 60	sselid 3992	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ On)
62		ontri1 6419	. . . . . . . 8 ⊢ ((𝑛 ∈ On ∧ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ On) → (𝑛 ⊆ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ↔ ¬ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ 𝑛))
63	45, 61, 62	syl2anc 584	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → (𝑛 ⊆ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ↔ ¬ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ 𝑛))
64		ssintrab 4975	. . . . . . . 8 ⊢ (𝑛 ⊆ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ↔ ∀𝑧 ∈ ω (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧))
65		nnon 7892	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
66		ontri1 6419	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ On ∧ 𝑧 ∈ On) → (𝑛 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑛))
67	44, 65, 66	syl2an 596	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → (𝑛 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑛))
68	67	imbi2d 340	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧) ↔ (¬ (𝐷‘𝑧) ∈ 𝑥 → ¬ 𝑧 ∈ 𝑛)))
69		con34b 316	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥) ↔ (¬ (𝐷‘𝑧) ∈ 𝑥 → ¬ 𝑧 ∈ 𝑛))
70	68, 69	bitr4di 289	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧) ↔ (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)))
71	70	pm5.74da 804	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) ↔ (𝑧 ∈ ω → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥))))
72		bi2.04 387	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)) ↔ (𝑧 ∈ 𝑛 → (𝑧 ∈ ω → (𝐷‘𝑧) ∈ 𝑥)))
73	71, 72	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) ↔ (𝑧 ∈ 𝑛 → (𝑧 ∈ ω → (𝐷‘𝑧) ∈ 𝑥))))
74		elnn 7897	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑛 ∧ 𝑛 ∈ ω) → 𝑧 ∈ ω)
75		pm2.27 42	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ ω → ((𝑧 ∈ ω → (𝐷‘𝑧) ∈ 𝑥) → (𝐷‘𝑧) ∈ 𝑥))
76	74, 75	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝑛 ∧ 𝑛 ∈ ω) → ((𝑧 ∈ ω → (𝐷‘𝑧) ∈ 𝑥) → (𝐷‘𝑧) ∈ 𝑥))
77	76	expcom 413	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → (𝑧 ∈ 𝑛 → ((𝑧 ∈ ω → (𝐷‘𝑧) ∈ 𝑥) → (𝐷‘𝑧) ∈ 𝑥)))
78	77	a2d 29	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ((𝑧 ∈ 𝑛 → (𝑧 ∈ ω → (𝐷‘𝑧) ∈ 𝑥)) → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)))
79	73, 78	sylbid 240	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)))
80	79	ad2antrl 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)))
81	80	ralimdv2 3160	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → (∀𝑧 ∈ ω (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧) → ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
82	64, 81	biimtrid 242	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → (𝑛 ⊆ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
83	63, 82	sylbird 260	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → (¬ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ 𝑛 → ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
84	43, 83	mt3d 148	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ 𝑛)
85	30, 84	ffvelcdmd 7104	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝐴)
86		fveq2 6906	. . . . . . . . 9 ⊢ (𝑦 = ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → (𝐷‘𝑦) = (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))
87	86	eleq1d 2823	. . . . . . . 8 ⊢ (𝑦 = ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → ((𝐷‘𝑦) ∈ 𝑥 ↔ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝑥))
88	87	notbid 318	. . . . . . 7 ⊢ (𝑦 = ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → (¬ (𝐷‘𝑦) ∈ 𝑥 ↔ ¬ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝑥))
89		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝐷‘𝑧) = (𝐷‘𝑦))
90	89	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝐷‘𝑧) ∈ 𝑥 ↔ (𝐷‘𝑦) ∈ 𝑥))
91	90	notbid 318	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (¬ (𝐷‘𝑧) ∈ 𝑥 ↔ ¬ (𝐷‘𝑦) ∈ 𝑥))
92	91	cbvrabv 3443	. . . . . . 7 ⊢ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} = {𝑦 ∈ ω ∣ ¬ (𝐷‘𝑦) ∈ 𝑥}
93	88, 92	elrab2 3697	. . . . . 6 ⊢ (∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ↔ (∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ ω ∧ ¬ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝑥))
94	93	simprbi 496	. . . . 5 ⊢ (∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → ¬ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝑥)
95	60, 94	syl 17	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → ¬ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝑥)
96	85, 95	eldifd 3973	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑m 𝑛))) → (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ (𝐴 ∖ 𝑥))
97	28, 96	rexlimddv 3158	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ (𝐴 ∖ 𝑥))
98	17, 97	eqeltrd 2838	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536   ∈ wcel 2105   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  {crab 3432  Vcvv 3477   ∖ cdif 3959   ⊆ wss 3962  ∅c0 4338  ifcif 4530  𝒫 cpw 4604  ∩ cint 4950  ∪ ciun 4995   class class class wbr 5147   We wwe 5639   × cxp 5686  ^◡ccnv 5687  ran crn 5689  Ord word 6384  Oncon0 6385   Fn wfn 6557  ⟶wf 6558  –1-1→wf1 6559  –1-1-onto→wf1o 6561  ‘cfv 6562  (class class class)co 7430   ∈ cmpo 7432  ωcom 7886   ↑_m cmap 8864   ≼ cdom 8981   ≺ csdm 8982  Fincfn 8983  cardccrd 9972

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987

This theorem is referenced by:  pwfseqlem4a  10698  pwfseqlem4  10699