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Theorem pwfseqlem3 9739
Description: Lemma for pwfseq 9743. Using the construction 𝐷 from pwfseqlem1 9737, 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 3353 . . . 4 𝑥 ∈ V
2 vex 3353 . . . 4 𝑟 ∈ V
3 fvex 6392 . . . . 5 (𝐻‘(card‘𝑥)) ∈ V
4 fvex 6392 . . . . 5 (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ V
53, 4ifex 4293 . . . 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 6985 . . . 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 1585 . . 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 490 . . . . . . 7 (𝜓 → ω ≼ 𝑥)
1110adantl 473 . . . . . 6 ((𝜑𝜓) → ω ≼ 𝑥)
12 domnsym 8297 . . . . . 6 (ω ≼ 𝑥 → ¬ 𝑥 ≺ ω)
1311, 12syl 17 . . . . 5 ((𝜑𝜓) → ¬ 𝑥 ≺ ω)
14 isfinite 8768 . . . . 5 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
1513, 14sylnibr 320 . . . 4 ((𝜑𝜓) → ¬ 𝑥 ∈ Fin)
1615iffalsed 4256 . . 3 ((𝜑𝜓) → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) = (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
178, 16syl5eq 2811 . 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 9737 . . . . . 6 ((𝜑𝜓) → 𝐷 ∈ ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∖ 𝑛 ∈ ω (𝑥𝑚 𝑛)))
24 eldif 3744 . . . . . 6 (𝐷 ∈ ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∖ 𝑛 ∈ ω (𝑥𝑚 𝑛)) ↔ (𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)))
2523, 24sylib 209 . . . . 5 ((𝜑𝜓) → (𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)))
2625simpld 488 . . . 4 ((𝜑𝜓) → 𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛))
27 eliun 4682 . . . 4 (𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛) ↔ ∃𝑛 ∈ ω 𝐷 ∈ (𝐴𝑚 𝑛))
2826, 27sylib 209 . . 3 ((𝜑𝜓) → ∃𝑛 ∈ ω 𝐷 ∈ (𝐴𝑚 𝑛))
29 elmapi 8086 . . . . . 6 (𝐷 ∈ (𝐴𝑚 𝑛) → 𝐷:𝑛𝐴)
3029ad2antll 720 . . . . 5 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝐷:𝑛𝐴)
31 ssiun2 4721 . . . . . . . . 9 (𝑛 ∈ ω → (𝑥𝑚 𝑛) ⊆ 𝑛 ∈ ω (𝑥𝑚 𝑛))
3231ad2antrl 719 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝑥𝑚 𝑛) ⊆ 𝑛 ∈ ω (𝑥𝑚 𝑛))
3325simprd 489 . . . . . . . . 9 ((𝜑𝜓) → ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛))
3433adantr 472 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛))
3532, 34ssneldd 3766 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ 𝐷 ∈ (𝑥𝑚 𝑛))
36 vex 3353 . . . . . . . . 9 𝑛 ∈ V
371, 36elmap 8093 . . . . . . . 8 (𝐷 ∈ (𝑥𝑚 𝑛) ↔ 𝐷:𝑛𝑥)
38 ffn 6225 . . . . . . . . 9 (𝐷:𝑛𝐴𝐷 Fn 𝑛)
39 ffnfv 6582 . . . . . . . . . 10 (𝐷:𝑛𝑥 ↔ (𝐷 Fn 𝑛 ∧ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
4039baib 531 . . . . . . . . 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 274 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷 ∈ (𝑥𝑚 𝑛) ↔ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
4335, 42mtbid 315 . . . . . 6 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥)
44 nnon 7273 . . . . . . . . 9 (𝑛 ∈ ω → 𝑛 ∈ On)
4544ad2antrl 719 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ On)
46 ssrab2 3849 . . . . . . . . . 10 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ ω
47 omsson 7271 . . . . . . . . . 10 ω ⊆ On
4846, 47sstri 3772 . . . . . . . . 9 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ On
49 ordom 7276 . . . . . . . . . . . . 13 Ord ω
50 simprl 787 . . . . . . . . . . . . 13 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ ω)
51 ordelss 5926 . . . . . . . . . . . . 13 ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
5249, 50, 51sylancr 581 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ⊆ ω)
53 rexnal 3141 . . . . . . . . . . . . 13 (∃𝑧𝑛 ¬ (𝐷𝑧) ∈ 𝑥 ↔ ¬ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥)
5443, 53sylibr 225 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ∃𝑧𝑛 ¬ (𝐷𝑧) ∈ 𝑥)
55 ssrexv 3829 . . . . . . . . . . . 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 4124 . . . . . . . . . . 11 ({𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅ ↔ ∃𝑧 ∈ ω ¬ (𝐷𝑧) ∈ 𝑥)
5856, 57sylibr 225 . . . . . . . . . 10 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅)
59 onint 7197 . . . . . . . . . 10 (({𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ On ∧ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})
6048, 58, 59sylancr 581 . . . . . . . . 9 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})
6148, 60sseldi 3761 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ On)
62 ontri1 5944 . . . . . . . 8 ((𝑛 ∈ On ∧ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ On) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛))
6345, 61, 62syl2anc 579 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛))
64 ssintrab 4658 . . . . . . . 8 (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ∀𝑧 ∈ ω (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧))
65 nnon 7273 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ω → 𝑧 ∈ On)
66 ontri1 5944 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ On ∧ 𝑧 ∈ On) → (𝑛𝑧 ↔ ¬ 𝑧𝑛))
6744, 65, 66syl2an 589 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → (𝑛𝑧 ↔ ¬ 𝑧𝑛))
6867imbi2d 331 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) ↔ (¬ (𝐷𝑧) ∈ 𝑥 → ¬ 𝑧𝑛)))
69 con34b 307 . . . . . . . . . . . . . 14 ((𝑧𝑛 → (𝐷𝑧) ∈ 𝑥) ↔ (¬ (𝐷𝑧) ∈ 𝑥 → ¬ 𝑧𝑛))
7068, 69syl6bbr 280 . . . . . . . . . . . . 13 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) ↔ (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
7170pm5.74da 838 . . . . . . . . . . . 12 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) ↔ (𝑧 ∈ ω → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥))))
72 bi2.04 377 . . . . . . . . . . . 12 ((𝑧 ∈ ω → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)) ↔ (𝑧𝑛 → (𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥)))
7371, 72syl6bb 278 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) ↔ (𝑧𝑛 → (𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥))))
74 elnn 7277 . . . . . . . . . . . . . 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 402 . . . . . . . . . . . 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 231 . . . . . . . . . 10 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
8079ad2antrl 719 . . . . . . . . 9 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
8180ralimdv2 3108 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (∀𝑧 ∈ ω (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8264, 81syl5bi 233 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8363, 82sylbird 251 . . . . . 6 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛 → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8443, 83mt3d 142 . . . . 5 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛)
8530, 84ffvelrnd 6554 . . . 4 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝐴)
86 fveq2 6379 . . . . . . . . 9 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → (𝐷𝑦) = (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
8786eleq1d 2829 . . . . . . . 8 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → ((𝐷𝑦) ∈ 𝑥 ↔ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
8887notbid 309 . . . . . . 7 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → (¬ (𝐷𝑦) ∈ 𝑥 ↔ ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
89 fveq2 6379 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝐷𝑧) = (𝐷𝑦))
9089eleq1d 2829 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝐷𝑧) ∈ 𝑥 ↔ (𝐷𝑦) ∈ 𝑥))
9190notbid 309 . . . . . . . 8 (𝑧 = 𝑦 → (¬ (𝐷𝑧) ∈ 𝑥 ↔ ¬ (𝐷𝑦) ∈ 𝑥))
9291cbvrabv 3348 . . . . . . 7 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} = {𝑦 ∈ ω ∣ ¬ (𝐷𝑦) ∈ 𝑥}
9388, 92elrab2 3525 . . . . . 6 ( {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ( {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ ω ∧ ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
9493simprbi 490 . . . . 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 3745 . . 3 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ (𝐴𝑥))
9728, 96rexlimddv 3182 . 2 ((𝜑𝜓) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ (𝐴𝑥))
9817, 97eqeltrd 2844 1 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  cdif 3731  wss 3734  c0 4081  ifcif 4245  𝒫 cpw 4317   cint 4635   ciun 4678   class class class wbr 4811   We wwe 5237   × cxp 5277  ccnv 5278  ran crn 5280  Ord word 5909  Oncon0 5910   Fn wfn 6065  wf 6066  1-1wf1 6067  1-1-ontowf1o 6069  cfv 6070  (class class class)co 6846  cmpt2 6848  ωcom 7267  𝑚 cmap 8064  cdom 8162  csdm 8163  Fincfn 8164  cardccrd 9016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-inf2 8757
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-1st 7370  df-2nd 7371  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-er 7951  df-map 8066  df-en 8165  df-dom 8166  df-sdom 8167  df-fin 8168
This theorem is referenced by:  pwfseqlem4a  9740  pwfseqlem4  9741
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