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Theorem pwfseqlem3 10651

Description: Lemma for pwfseq 10655. Using the construction 𝐷 from pwfseqlem1 10649, 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 3478	. . . 4 ⊢ 𝑥 ∈ V
2		vex 3478	. . . 4 ⊢ 𝑟 ∈ V
3		fvex 6901	. . . . 5 ⊢ (𝐻‘(card‘𝑥)) ∈ V
4		fvex 6901	. . . . 5 ⊢ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ V
5	3, 4	ifex 4577	. . . 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 7551	. . . 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 1461	. . 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 497	. . . . . . 7 ⊢ (𝜓 → ω ≼ 𝑥)
11	10	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ω ≼ 𝑥)
12		domnsym 9095	. . . . . 6 ⊢ (ω ≼ 𝑥 → ¬ 𝑥 ≺ ω)
13	11, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑥 ≺ ω)
14		isfinite 9643	. . . . 5 ⊢ (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
15	13, 14	sylnibr 328	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑥 ∈ Fin)
16	15	iffalsed 4538	. . 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 2784	. 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 10649	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ (∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∖ ∪ 𝑛 ∈ ω (𝑥 ↑_m 𝑛)))
24		eldif 3957	. . . . . 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 217	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ∧ ¬ 𝐷 ∈ ∪ 𝑛 ∈ ω (𝑥 ↑_m 𝑛)))
26	25	simpld 495	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → 𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛))
27		eliun 5000	. . . 4 ⊢ (𝐷 ∈ ∪ 𝑛 ∈ ω (𝐴 ↑_m 𝑛) ↔ ∃𝑛 ∈ ω 𝐷 ∈ (𝐴 ↑_m 𝑛))
28	26, 27	sylib 217	. . 3 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑛 ∈ ω 𝐷 ∈ (𝐴 ↑_m 𝑛))
29		elmapi 8839	. . . . . 6 ⊢ (𝐷 ∈ (𝐴 ↑_m 𝑛) → 𝐷:𝑛⟶𝐴)
30	29	ad2antll 727	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → 𝐷:𝑛⟶𝐴)
31		ssiun2 5049	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → (𝑥 ↑_m 𝑛) ⊆ ∪ 𝑛 ∈ ω (𝑥 ↑_m 𝑛))
32	31	ad2antrl 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → (𝑥 ↑_m 𝑛) ⊆ ∪ 𝑛 ∈ ω (𝑥 ↑_m 𝑛))
33	25	simprd 496	. . . . . . . . 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 3984	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → ¬ 𝐷 ∈ (𝑥 ↑_m 𝑛))
36		vex 3478	. . . . . . . . 9 ⊢ 𝑛 ∈ V
37	1, 36	elmap 8861	. . . . . . . 8 ⊢ (𝐷 ∈ (𝑥 ↑_m 𝑛) ↔ 𝐷:𝑛⟶𝑥)
38		ffn 6714	. . . . . . . . 9 ⊢ (𝐷:𝑛⟶𝐴 → 𝐷 Fn 𝑛)
39		ffnfv 7114	. . . . . . . . . 10 ⊢ (𝐷:𝑛⟶𝑥 ↔ (𝐷 Fn 𝑛 ∧ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
40	39	baib 536	. . . . . . . . 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 282	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → (𝐷 ∈ (𝑥 ↑_m 𝑛) ↔ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
43	35, 42	mtbid 323	. . . . . 6 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → ¬ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥)
44		nnon 7857	. . . . . . . . 9 ⊢ (𝑛 ∈ ω → 𝑛 ∈ On)
45	44	ad2antrl 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → 𝑛 ∈ On)
46		ssrab2 4076	. . . . . . . . . 10 ⊢ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ⊆ ω
47		omsson 7855	. . . . . . . . . 10 ⊢ ω ⊆ On
48	46, 47	sstri 3990	. . . . . . . . 9 ⊢ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ⊆ On
49		ordom 7861	. . . . . . . . . . . . 13 ⊢ Ord ω
50		simprl 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → 𝑛 ∈ ω)
51		ordelss 6377	. . . . . . . . . . . . 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 3100	. . . . . . . . . . . . 13 ⊢ (∃𝑧 ∈ 𝑛 ¬ (𝐷‘𝑧) ∈ 𝑥 ↔ ¬ ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥)
54	43, 53	sylibr 233	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → ∃𝑧 ∈ 𝑛 ¬ (𝐷‘𝑧) ∈ 𝑥)
55		ssrexv 4050	. . . . . . . . . . . 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 4384	. . . . . . . . . . 11 ⊢ ({𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ≠ ∅ ↔ ∃𝑧 ∈ ω ¬ (𝐷‘𝑧) ∈ 𝑥)
58	56, 57	sylibr 233	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ≠ ∅)
59		onint 7774	. . . . . . . . . 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 3979	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ∈ On)
62		ontri1 6395	. . . . . . . 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 4974	. . . . . . . 8 ⊢ (𝑛 ⊆ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} ↔ ∀𝑧 ∈ ω (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧))
65		nnon 7857	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 ∈ ω → 𝑧 ∈ On)
66		ontri1 6395	. . . . . . . . . . . . . . . 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 315	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥) ↔ (¬ (𝐷‘𝑧) ∈ 𝑥 → ¬ 𝑧 ∈ 𝑛))
70	68, 69	bitr4di 288	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧) ↔ (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)))
71	70	pm5.74da 802	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) ↔ (𝑧 ∈ ω → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥))))
72		bi2.04 388	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ ω → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)) ↔ (𝑧 ∈ 𝑛 → (𝑧 ∈ ω → (𝐷‘𝑧) ∈ 𝑥)))
73	71, 72	bitrdi 286	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) ↔ (𝑧 ∈ 𝑛 → (𝑧 ∈ ω → (𝐷‘𝑧) ∈ 𝑥))))
74		elnn 7862	. . . . . . . . . . . . . 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 414	. . . . . . . . . . . 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 239	. . . . . . . . . 10 ⊢ (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)))
80	79	ad2antrl 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → ((𝑧 ∈ ω → (¬ (𝐷‘𝑧) ∈ 𝑥 → 𝑛 ⊆ 𝑧)) → (𝑧 ∈ 𝑛 → (𝐷‘𝑧) ∈ 𝑥)))
81	80	ralimdv2 3163	. . . . . . . 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 241	. . . . . . 7 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → (𝑛 ⊆ ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → ∀𝑧 ∈ 𝑛 (𝐷‘𝑧) ∈ 𝑥))
83	63, 82	sylbird 259	. . . . . 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 7084	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝐴)
86		fveq2 6888	. . . . . . . . 9 ⊢ (𝑦 = ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → (𝐷‘𝑦) = (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}))
87	86	eleq1d 2818	. . . . . . . 8 ⊢ (𝑦 = ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → ((𝐷‘𝑦) ∈ 𝑥 ↔ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝑥))
88	87	notbid 317	. . . . . . 7 ⊢ (𝑦 = ∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} → (¬ (𝐷‘𝑦) ∈ 𝑥 ↔ ¬ (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ 𝑥))
89		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝐷‘𝑧) = (𝐷‘𝑦))
90	89	eleq1d 2818	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((𝐷‘𝑧) ∈ 𝑥 ↔ (𝐷‘𝑦) ∈ 𝑥))
91	90	notbid 317	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (¬ (𝐷‘𝑧) ∈ 𝑥 ↔ ¬ (𝐷‘𝑦) ∈ 𝑥))
92	91	cbvrabv 3442	. . . . . . 7 ⊢ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥} = {𝑦 ∈ ω ∣ ¬ (𝐷‘𝑦) ∈ 𝑥}
93	88, 92	elrab2 3685	. . . . . 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 497	. . . . 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 3958	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴 ↑_m 𝑛))) → (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ (𝐴 ∖ 𝑥))
97	28, 96	rexlimddv 3161	. 2 ⊢ ((𝜑 ∧ 𝜓) → (𝐷‘∩ {𝑧 ∈ ω ∣ ¬ (𝐷‘𝑧) ∈ 𝑥}) ∈ (𝐴 ∖ 𝑥))
98	17, 97	eqeltrd 2833	1 ⊢ ((𝜑 ∧ 𝜓) → (𝑥𝐹𝑟) ∈ (𝐴 ∖ 𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 ifcif 4527 𝒫 cpw 4601 ∩ cint 4949 ∪ ciun 4996 class class class wbr 5147 We wwe 5629 × cxp 5673 ^◡ccnv 5674 ran crn 5676 Ord word 6360 Oncon0 6361 Fn wfn 6535 ⟶wf 6536 –1-1→wf1 6537 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 ωcom 7851 ↑_m cmap 8816 ≼ cdom 8933 ≺ csdm 8934 Fincfn 8935 cardccrd 9926

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939

This theorem is referenced by: pwfseqlem4a 10652 pwfseqlem4 10653

W3C validator