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Theorem pwfseqlem3 9882
Description: Lemma for pwfseq 9886. Using the construction 𝐷 from pwfseqlem1 9880, 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 (𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
pwfseqlem4.x (𝜑𝑋𝐴)
pwfseqlem4.h (𝜑𝐻:ω–1-1-onto𝑋)
pwfseqlem4.ps (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
pwfseqlem4.k ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)
pwfseqlem4.d 𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
pwfseqlem4.f 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
Assertion
Ref Expression
pwfseqlem3 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
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.
StepHypRef Expression
1 vex 3418 . . . 4 𝑥 ∈ V
2 vex 3418 . . . 4 𝑟 ∈ V
3 fvex 6514 . . . . 5 (𝐻‘(card‘𝑥)) ∈ V
4 fvex 6514 . . . . 5 (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ V
53, 4ifex 4399 . . . 4 if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) ∈ V
6 pwfseqlem4.f . . . . 5 𝐹 = (𝑥 ∈ V, 𝑟 ∈ V ↦ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
76ovmpt4g 7115 . . . 4 ((𝑥 ∈ V ∧ 𝑟 ∈ V ∧ if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) ∈ V) → (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})))
81, 2, 5, 7mp3an 1440 . . 3 (𝑥𝐹𝑟) = if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
9 pwfseqlem4.ps . . . . . . . 8 (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
109simprbi 489 . . . . . . 7 (𝜓 → ω ≼ 𝑥)
1110adantl 474 . . . . . 6 ((𝜑𝜓) → ω ≼ 𝑥)
12 domnsym 8441 . . . . . 6 (ω ≼ 𝑥 → ¬ 𝑥 ≺ ω)
1311, 12syl 17 . . . . 5 ((𝜑𝜓) → ¬ 𝑥 ≺ ω)
14 isfinite 8911 . . . . 5 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
1513, 14sylnibr 321 . . . 4 ((𝜑𝜓) → ¬ 𝑥 ∈ Fin)
1615iffalsed 4362 . . 3 ((𝜑𝜓) → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) = (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
178, 16syl5eq 2826 . 2 ((𝜑𝜓) → (𝑥𝐹𝑟) = (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
18 pwfseqlem4.g . . . . . . 7 (𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
19 pwfseqlem4.x . . . . . . 7 (𝜑𝑋𝐴)
20 pwfseqlem4.h . . . . . . 7 (𝜑𝐻:ω–1-1-onto𝑋)
21 pwfseqlem4.k . . . . . . 7 ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)
22 pwfseqlem4.d . . . . . . 7 𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
2318, 19, 20, 9, 21, 22pwfseqlem1 9880 . . . . . 6 ((𝜑𝜓) → 𝐷 ∈ ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∖ 𝑛 ∈ ω (𝑥𝑚 𝑛)))
24 eldif 3841 . . . . . 6 (𝐷 ∈ ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∖ 𝑛 ∈ ω (𝑥𝑚 𝑛)) ↔ (𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)))
2523, 24sylib 210 . . . . 5 ((𝜑𝜓) → (𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)))
2625simpld 487 . . . 4 ((𝜑𝜓) → 𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛))
27 eliun 4797 . . . 4 (𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛) ↔ ∃𝑛 ∈ ω 𝐷 ∈ (𝐴𝑚 𝑛))
2826, 27sylib 210 . . 3 ((𝜑𝜓) → ∃𝑛 ∈ ω 𝐷 ∈ (𝐴𝑚 𝑛))
29 elmapi 8230 . . . . . 6 (𝐷 ∈ (𝐴𝑚 𝑛) → 𝐷:𝑛𝐴)
3029ad2antll 716 . . . . 5 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝐷:𝑛𝐴)
31 ssiun2 4838 . . . . . . . . 9 (𝑛 ∈ ω → (𝑥𝑚 𝑛) ⊆ 𝑛 ∈ ω (𝑥𝑚 𝑛))
3231ad2antrl 715 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝑥𝑚 𝑛) ⊆ 𝑛 ∈ ω (𝑥𝑚 𝑛))
3325simprd 488 . . . . . . . . 9 ((𝜑𝜓) → ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛))
3433adantr 473 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛))
3532, 34ssneldd 3863 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ 𝐷 ∈ (𝑥𝑚 𝑛))
36 vex 3418 . . . . . . . . 9 𝑛 ∈ V
371, 36elmap 8237 . . . . . . . 8 (𝐷 ∈ (𝑥𝑚 𝑛) ↔ 𝐷:𝑛𝑥)
38 ffn 6346 . . . . . . . . 9 (𝐷:𝑛𝐴𝐷 Fn 𝑛)
39 ffnfv 6707 . . . . . . . . . 10 (𝐷:𝑛𝑥 ↔ (𝐷 Fn 𝑛 ∧ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
4039baib 528 . . . . . . . . 9 (𝐷 Fn 𝑛 → (𝐷:𝑛𝑥 ↔ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
4130, 38, 403syl 18 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷:𝑛𝑥 ↔ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
4237, 41syl5bb 275 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷 ∈ (𝑥𝑚 𝑛) ↔ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
4335, 42mtbid 316 . . . . . 6 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥)
44 nnon 7404 . . . . . . . . 9 (𝑛 ∈ ω → 𝑛 ∈ On)
4544ad2antrl 715 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ On)
46 ssrab2 3948 . . . . . . . . . 10 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ ω
47 omsson 7402 . . . . . . . . . 10 ω ⊆ On
4846, 47sstri 3869 . . . . . . . . 9 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ On
49 ordom 7407 . . . . . . . . . . . . 13 Ord ω
50 simprl 758 . . . . . . . . . . . . 13 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ ω)
51 ordelss 6047 . . . . . . . . . . . . 13 ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
5249, 50, 51sylancr 578 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ⊆ ω)
53 rexnal 3185 . . . . . . . . . . . . 13 (∃𝑧𝑛 ¬ (𝐷𝑧) ∈ 𝑥 ↔ ¬ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥)
5443, 53sylibr 226 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ∃𝑧𝑛 ¬ (𝐷𝑧) ∈ 𝑥)
55 ssrexv 3926 . . . . . . . . . . . 12 (𝑛 ⊆ ω → (∃𝑧𝑛 ¬ (𝐷𝑧) ∈ 𝑥 → ∃𝑧 ∈ ω ¬ (𝐷𝑧) ∈ 𝑥))
5652, 54, 55sylc 65 . . . . . . . . . . 11 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ∃𝑧 ∈ ω ¬ (𝐷𝑧) ∈ 𝑥)
57 rabn0 4227 . . . . . . . . . . 11 ({𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅ ↔ ∃𝑧 ∈ ω ¬ (𝐷𝑧) ∈ 𝑥)
5856, 57sylibr 226 . . . . . . . . . 10 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅)
59 onint 7328 . . . . . . . . . 10 (({𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ On ∧ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})
6048, 58, 59sylancr 578 . . . . . . . . 9 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})
6148, 60sseldi 3858 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ On)
62 ontri1 6065 . . . . . . . 8 ((𝑛 ∈ On ∧ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ On) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛))
6345, 61, 62syl2anc 576 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛))
64 ssintrab 4773 . . . . . . . 8 (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ∀𝑧 ∈ ω (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧))
65 nnon 7404 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ω → 𝑧 ∈ On)
66 ontri1 6065 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ On ∧ 𝑧 ∈ On) → (𝑛𝑧 ↔ ¬ 𝑧𝑛))
6744, 65, 66syl2an 586 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → (𝑛𝑧 ↔ ¬ 𝑧𝑛))
6867imbi2d 333 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) ↔ (¬ (𝐷𝑧) ∈ 𝑥 → ¬ 𝑧𝑛)))
69 con34b 308 . . . . . . . . . . . . . 14 ((𝑧𝑛 → (𝐷𝑧) ∈ 𝑥) ↔ (¬ (𝐷𝑧) ∈ 𝑥 → ¬ 𝑧𝑛))
7068, 69syl6bbr 281 . . . . . . . . . . . . 13 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) ↔ (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
7170pm5.74da 791 . . . . . . . . . . . 12 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) ↔ (𝑧 ∈ ω → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥))))
72 bi2.04 380 . . . . . . . . . . . 12 ((𝑧 ∈ ω → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)) ↔ (𝑧𝑛 → (𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥)))
7371, 72syl6bb 279 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) ↔ (𝑧𝑛 → (𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥))))
74 elnn 7408 . . . . . . . . . . . . . 14 ((𝑧𝑛𝑛 ∈ ω) → 𝑧 ∈ ω)
75 pm2.27 42 . . . . . . . . . . . . . 14 (𝑧 ∈ ω → ((𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥) → (𝐷𝑧) ∈ 𝑥))
7674, 75syl 17 . . . . . . . . . . . . 13 ((𝑧𝑛𝑛 ∈ ω) → ((𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥) → (𝐷𝑧) ∈ 𝑥))
7776expcom 406 . . . . . . . . . . . 12 (𝑛 ∈ ω → (𝑧𝑛 → ((𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥) → (𝐷𝑧) ∈ 𝑥)))
7877a2d 29 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑧𝑛 → (𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥)) → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
7973, 78sylbid 232 . . . . . . . . . 10 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
8079ad2antrl 715 . . . . . . . . 9 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
8180ralimdv2 3126 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (∀𝑧 ∈ ω (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8264, 81syl5bi 234 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8363, 82sylbird 252 . . . . . 6 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛 → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8443, 83mt3d 143 . . . . 5 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛)
8530, 84ffvelrnd 6679 . . . 4 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝐴)
86 fveq2 6501 . . . . . . . . 9 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → (𝐷𝑦) = (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
8786eleq1d 2850 . . . . . . . 8 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → ((𝐷𝑦) ∈ 𝑥 ↔ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
8887notbid 310 . . . . . . 7 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → (¬ (𝐷𝑦) ∈ 𝑥 ↔ ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
89 fveq2 6501 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝐷𝑧) = (𝐷𝑦))
9089eleq1d 2850 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝐷𝑧) ∈ 𝑥 ↔ (𝐷𝑦) ∈ 𝑥))
9190notbid 310 . . . . . . . 8 (𝑧 = 𝑦 → (¬ (𝐷𝑧) ∈ 𝑥 ↔ ¬ (𝐷𝑦) ∈ 𝑥))
9291cbvrabv 3412 . . . . . . 7 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} = {𝑦 ∈ ω ∣ ¬ (𝐷𝑦) ∈ 𝑥}
9388, 92elrab2 3599 . . . . . 6 ( {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ( {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ ω ∧ ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
9493simprbi 489 . . . . 5 ( {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥)
9560, 94syl 17 . . . 4 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥)
9685, 95eldifd 3842 . . 3 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ (𝐴𝑥))
9728, 96rexlimddv 3236 . 2 ((𝜑𝜓) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ (𝐴𝑥))
9817, 97eqeltrd 2866 1 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2967  wral 3088  wrex 3089  {crab 3092  Vcvv 3415  cdif 3828  wss 3831  c0 4180  ifcif 4351  𝒫 cpw 4423   cint 4750   ciun 4793   class class class wbr 4930   We wwe 5366   × cxp 5406  ccnv 5407  ran crn 5409  Ord word 6030  Oncon0 6031   Fn wfn 6185  wf 6186  1-1wf1 6187  1-1-ontowf1o 6189  cfv 6190  (class class class)co 6978  cmpo 6980  ωcom 7398  𝑚 cmap 8208  cdom 8306  csdm 8307  Fincfn 8308  cardccrd 9160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281  ax-inf2 8900
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-1st 7503  df-2nd 7504  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-er 8091  df-map 8210  df-en 8309  df-dom 8310  df-sdom 8311  df-fin 8312
This theorem is referenced by:  pwfseqlem4a  9883  pwfseqlem4  9884
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