Theorem pwfseqlem3 9482

Description: Lemma for pwfseq 9486. Using the construction 𝐷 from pwfseqlem1 9480, 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 3203	. . . 4 ⊢ 𝑥 ∈ V
2		vex 3203	. . . 4 ⊢ 𝑟 ∈ V
3		fvex 6201	. . . . 5 ⊢ (𝐻‘(card‘𝑥)) ∈ V
4		fvex 6201	. . . . 5 ⊢ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ V
5	3, 4	ifex 4156	. . . 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 6783	. . . 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 1424	. . 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 480	. . . . . . 7 ⊢ (𝜓 → ω ≼ 𝑥)
11	10	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ 𝑥)
12		domnsym 8086	. . . . . 6 ⊢ (ω ≼ 𝑥 → ¬ 𝑥 ≺ ω)
13	11, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑥 ≺ ω)
14		isfinite 8549	. . . . 5 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
15	13, 14	sylnibr 319	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑥 ∈ Fin)
16	15	iffalsed 4097	. . 3 ⊢ ((𝜑 ∧ 𝜓) → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})) = (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))
17	8, 16	syl5eq 2668	. 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 9480	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ (∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∖ ∪ 𝑛 ∈ ω (𝑥 ↑𝑚 𝑛)))
24		eldif 3584	. . . . . 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 208	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ∧ ¬ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑𝑚 𝑛)))
26	25	simpld 475	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛))
27		eliun 4524	. . . 4 ⊢ (𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑𝑚 𝑛) ↔ ∃𝑛 ∈ ω 𝐷 ∈ (𝐴 ↑𝑚 𝑛))
28	26, 27	sylib 208	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑛 ∈ ω 𝐷 ∈ (𝐴 ↑𝑚 𝑛))
29		elmapi 7879	. . . . . 6 ⊢ (𝐷 ∈ (𝐴 ↑𝑚 𝑛) → 𝐷:𝑛⟶𝐴)
30	29	ad2antll 765	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → 𝐷:𝑛⟶𝐴)
31		ssiun2 4563	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → (𝑥 ↑𝑚 𝑛) ⊆ ∪ 𝑛 ∈ ω (𝑥 ↑𝑚 𝑛))
32	31	ad2antrl 764	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → (𝑥 ↑𝑚 𝑛) ⊆ ∪ 𝑛 ∈ ω (𝑥 ↑𝑚 𝑛))
33	25	simprd 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑𝑚 𝑛))
34	33	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → ¬ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑𝑚 𝑛))
35	32, 34	ssneldd 3606	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → ¬ 𝐷 ∈ (𝑥 ↑𝑚 𝑛))
36		vex 3203	. . . . . . . . 9 ⊢ 𝑛 ∈ V
37	1, 36	elmap 7886	. . . . . . . 8 ⊢ (𝐷 ∈ (𝑥 ↑𝑚 𝑛) ↔ 𝐷:𝑛⟶𝑥)
38		ffn 6045	. . . . . . . . 9 ⊢ (𝐷:𝑛⟶𝐴 → 𝐷 Fn 𝑛)
39		ffnfv 6388	. . . . . . . . . 10 ⊢ (𝐷:𝑛⟶𝑥 ↔ (𝐷 Fn 𝑛 ∧ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
40	39	baib 944	. . . . . . . . 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	syl5bb 272	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → (𝐷 ∈ (𝑥 ↑𝑚 𝑛) ↔ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
43	35, 42	mtbid 314	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → ¬ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥)
44		nnon 7071	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
45	44	ad2antrl 764	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → 𝑛 ∈ On)
46		ssrab2 3687	. . . . . . . . . 10 ⊢ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ⊆ ω
47		omsson 7069	. . . . . . . . . 10 ⊢ ω ⊆ On
48	46, 47	sstri 3612	. . . . . . . . 9 ⊢ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ⊆ On
49		ordom 7074	. . . . . . . . . . . . 13 ⊢ Ord ω
50		simprl 794	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → 𝑛 ∈ ω)
51		ordelss 5739	. . . . . . . . . . . . 13 ⊢ ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
52	49, 50, 51	sylancr 695	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → 𝑛 ⊆ ω)
53		rexnal 2995	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ 𝑛 ¬ (𝐷‘𝑧) ∈ 𝑥 ↔ ¬ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥)
54	43, 53	sylibr 224	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → ∃𝑧 ∈ 𝑛 ¬ (𝐷‘𝑧) ∈ 𝑥)
55		ssrexv 3667	. . . . . . . . . . . 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 3958	. . . . . . . . . . 11 ⊢ ({𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ≠ ∅ ↔ ∃𝑧 ∈ ω ¬ (𝐷‘𝑧) ∈ 𝑥)
58	56, 57	sylibr 224	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ≠ ∅)
59		onint 6995	. . . . . . . . . 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 695	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥})
61	48, 60	sseldi 3601	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ On)
62		ontri1 5757	. . . . . . . 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 693	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → (𝑛 ⊆ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ↔ ¬ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ 𝑛))
64		ssintrab 4500	. . . . . . . 8 ⊢ (𝑛 ⊆ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ↔ ∀𝑧 ∈ ω (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧))
65		nnon 7071	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
66		ontri1 5757	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ On ∧ 𝑧 ∈ On) → (𝑛 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑛))
67	44, 65, 66	syl2an 494	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → (𝑛 ⊆ 𝑧 ↔ ¬ 𝑧 ∈ 𝑛))
68	67	imbi2d 330	. . . . . . . . . . . . . 14 ⊢ ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧) ↔ (¬ (𝐷‘𝑧) ∈ 𝑥 → ¬ 𝑧 ∈ 𝑛)))
69		con34b 306	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥) ↔ (¬ (𝐷‘𝑧) ∈ 𝑥 → ¬ 𝑧 ∈ 𝑛))
70	68, 69	syl6bbr 278	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧) ↔ (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)))
71	70	pm5.74da 723	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) ↔ (𝑧 ∈ ω → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥))))
72		bi2.04 376	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)) ↔ (𝑧 ∈ 𝑛 → (𝑧 ∈ ω → (𝐷‘𝑧) ∈ 𝑥)))
73	71, 72	syl6bb 276	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) ↔ (𝑧 ∈ 𝑛 → (𝑧 ∈ ω → (𝐷‘𝑧) ∈ 𝑥))))
74		elnn 7075	. . . . . . . . . . . . . 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 451	. . . . . . . . . . . 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 230	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)))
80	79	ad2antrl 764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)))
81	80	ralimdv2 2961	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → (∀𝑧 ∈ ω (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧) → ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
82	64, 81	syl5bi 232	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → (𝑛 ⊆ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
83	63, 82	sylbird 250	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → (¬ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ 𝑛 → ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
84	43, 83	mt3d 140	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ 𝑛)
85	30, 84	ffvelrnd 6360	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝐴)
86		fveq2 6191	. . . . . . . . 9 ⊢ (𝑦 = ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → (𝐷‘𝑦) = (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))
87	86	eleq1d 2686	. . . . . . . 8 ⊢ (𝑦 = ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → ((𝐷‘𝑦) ∈ 𝑥 ↔ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝑥))
88	87	notbid 308	. . . . . . 7 ⊢ (𝑦 = ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → (¬ (𝐷‘𝑦) ∈ 𝑥 ↔ ¬ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝑥))
89		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝐷‘𝑧) = (𝐷‘𝑦))
90	89	eleq1d 2686	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝐷‘𝑧) ∈ 𝑥 ↔ (𝐷‘𝑦) ∈ 𝑥))
91	90	notbid 308	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (¬ (𝐷‘𝑧) ∈ 𝑥 ↔ ¬ (𝐷‘𝑦) ∈ 𝑥))
92	91	cbvrabv 3199	. . . . . . 7 ⊢ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} = {𝑦 ∈ ω ∣ ¬ (𝐷‘𝑦) ∈ 𝑥}
93	88, 92	elrab2 3366	. . . . . 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 480	. . . . 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 3585	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑𝑚 𝑛))) → (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ (𝐴 ∖ 𝑥))
97	28, 96	rexlimddv 3035	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ (𝐴 ∖ 𝑥))
98	17, 97	eqeltrd 2701	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ifcif 4086  𝒫 cpw 4158  ∩ cint 4475  ∪ ciun 4520   class class class wbr 4653   We wwe 5072   × cxp 5112  ^◡ccnv 5113  ran crn 5115  Ord word 5722  Oncon0 5723   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ωcom 7065   ↑_𝑚 cmap 7857   ≼ cdom 7953   ≺ csdm 7954  Fincfn 7955  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959

This theorem is referenced by:  pwfseqlem4a  9483  pwfseqlem4  9484