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Theorem pwfseqlem3 10691
Description: Lemma for pwfseq 10695. Using the construction 𝐷 from pwfseqlem1 10689, 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 3466 . . . 4 𝑥 ∈ V
2 vex 3466 . . . 4 𝑟 ∈ V
3 fvex 6903 . . . . 5 (𝐻‘(card‘𝑥)) ∈ V
4 fvex 6903 . . . . 5 (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ V
53, 4ifex 4573 . . . 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 7562 . . . 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 1458 . . 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 495 . . . . . . 7 (𝜓 → ω ≼ 𝑥)
1110adantl 480 . . . . . 6 ((𝜑𝜓) → ω ≼ 𝑥)
12 domnsym 9126 . . . . . 6 (ω ≼ 𝑥 → ¬ 𝑥 ≺ ω)
1311, 12syl 17 . . . . 5 ((𝜑𝜓) → ¬ 𝑥 ≺ ω)
14 isfinite 9685 . . . . 5 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
1513, 14sylnibr 328 . . . 4 ((𝜑𝜓) → ¬ 𝑥 ∈ Fin)
1615iffalsed 4534 . . 3 ((𝜑𝜓) → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) = (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
178, 16eqtrid 2778 . 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 10689 . . . . . 6 ((𝜑𝜓) → 𝐷 ∈ ( 𝑛 ∈ ω (𝐴m 𝑛) ∖ 𝑛 ∈ ω (𝑥m 𝑛)))
24 eldif 3956 . . . . . 6 (𝐷 ∈ ( 𝑛 ∈ ω (𝐴m 𝑛) ∖ 𝑛 ∈ ω (𝑥m 𝑛)) ↔ (𝐷 𝑛 ∈ ω (𝐴m 𝑛) ∧ ¬ 𝐷 𝑛 ∈ ω (𝑥m 𝑛)))
2523, 24sylib 217 . . . . 5 ((𝜑𝜓) → (𝐷 𝑛 ∈ ω (𝐴m 𝑛) ∧ ¬ 𝐷 𝑛 ∈ ω (𝑥m 𝑛)))
2625simpld 493 . . . 4 ((𝜑𝜓) → 𝐷 𝑛 ∈ ω (𝐴m 𝑛))
27 eliun 4997 . . . 4 (𝐷 𝑛 ∈ ω (𝐴m 𝑛) ↔ ∃𝑛 ∈ ω 𝐷 ∈ (𝐴m 𝑛))
2826, 27sylib 217 . . 3 ((𝜑𝜓) → ∃𝑛 ∈ ω 𝐷 ∈ (𝐴m 𝑛))
29 elmapi 8867 . . . . . 6 (𝐷 ∈ (𝐴m 𝑛) → 𝐷:𝑛𝐴)
3029ad2antll 727 . . . . 5 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → 𝐷:𝑛𝐴)
31 ssiun2 5047 . . . . . . . . 9 (𝑛 ∈ ω → (𝑥m 𝑛) ⊆ 𝑛 ∈ ω (𝑥m 𝑛))
3231ad2antrl 726 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → (𝑥m 𝑛) ⊆ 𝑛 ∈ ω (𝑥m 𝑛))
3325simprd 494 . . . . . . . . 9 ((𝜑𝜓) → ¬ 𝐷 𝑛 ∈ ω (𝑥m 𝑛))
3433adantr 479 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → ¬ 𝐷 𝑛 ∈ ω (𝑥m 𝑛))
3532, 34ssneldd 3981 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → ¬ 𝐷 ∈ (𝑥m 𝑛))
36 vex 3466 . . . . . . . . 9 𝑛 ∈ V
371, 36elmap 8889 . . . . . . . 8 (𝐷 ∈ (𝑥m 𝑛) ↔ 𝐷:𝑛𝑥)
38 ffn 6717 . . . . . . . . 9 (𝐷:𝑛𝐴𝐷 Fn 𝑛)
39 ffnfv 7122 . . . . . . . . . 10 (𝐷:𝑛𝑥 ↔ (𝐷 Fn 𝑛 ∧ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
4039baib 534 . . . . . . . . 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, 41bitrid 282 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → (𝐷 ∈ (𝑥m 𝑛) ↔ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
4335, 42mtbid 323 . . . . . 6 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → ¬ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥)
44 nnon 7871 . . . . . . . . 9 (𝑛 ∈ ω → 𝑛 ∈ On)
4544ad2antrl 726 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → 𝑛 ∈ On)
46 ssrab2 4073 . . . . . . . . . 10 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ ω
47 omsson 7869 . . . . . . . . . 10 ω ⊆ On
4846, 47sstri 3988 . . . . . . . . 9 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ On
49 ordom 7875 . . . . . . . . . . . . 13 Ord ω
50 simprl 769 . . . . . . . . . . . . 13 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → 𝑛 ∈ ω)
51 ordelss 6381 . . . . . . . . . . . . 13 ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
5249, 50, 51sylancr 585 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → 𝑛 ⊆ ω)
53 rexnal 3090 . . . . . . . . . . . . 13 (∃𝑧𝑛 ¬ (𝐷𝑧) ∈ 𝑥 ↔ ¬ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥)
5443, 53sylibr 233 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → ∃𝑧𝑛 ¬ (𝐷𝑧) ∈ 𝑥)
55 ssrexv 4048 . . . . . . . . . . . 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 4383 . . . . . . . . . . 11 ({𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅ ↔ ∃𝑧 ∈ ω ¬ (𝐷𝑧) ∈ 𝑥)
5856, 57sylibr 233 . . . . . . . . . 10 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅)
59 onint 7788 . . . . . . . . . 10 (({𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ On ∧ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})
6048, 58, 59sylancr 585 . . . . . . . . 9 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})
6148, 60sselid 3976 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ On)
62 ontri1 6399 . . . . . . . 8 ((𝑛 ∈ On ∧ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ On) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛))
6345, 61, 62syl2anc 582 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛))
64 ssintrab 4971 . . . . . . . 8 (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ∀𝑧 ∈ ω (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧))
65 nnon 7871 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ω → 𝑧 ∈ On)
66 ontri1 6399 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ On ∧ 𝑧 ∈ On) → (𝑛𝑧 ↔ ¬ 𝑧𝑛))
6744, 65, 66syl2an 594 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → (𝑛𝑧 ↔ ¬ 𝑧𝑛))
6867imbi2d 339 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) ↔ (¬ (𝐷𝑧) ∈ 𝑥 → ¬ 𝑧𝑛)))
69 con34b 315 . . . . . . . . . . . . . 14 ((𝑧𝑛 → (𝐷𝑧) ∈ 𝑥) ↔ (¬ (𝐷𝑧) ∈ 𝑥 → ¬ 𝑧𝑛))
7068, 69bitr4di 288 . . . . . . . . . . . . 13 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) ↔ (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
7170pm5.74da 802 . . . . . . . . . . . 12 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) ↔ (𝑧 ∈ ω → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥))))
72 bi2.04 386 . . . . . . . . . . . 12 ((𝑧 ∈ ω → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)) ↔ (𝑧𝑛 → (𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥)))
7371, 72bitrdi 286 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) ↔ (𝑧𝑛 → (𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥))))
74 elnn 7876 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . 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 239 . . . . . . . . . 10 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
8079ad2antrl 726 . . . . . . . . 9 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
8180ralimdv2 3153 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → (∀𝑧 ∈ ω (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8264, 81biimtrid 241 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8363, 82sylbird 259 . . . . . 6 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → (¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛 → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8443, 83mt3d 148 . . . . 5 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛)
8530, 84ffvelcdmd 7088 . . . 4 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝐴)
86 fveq2 6890 . . . . . . . . 9 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → (𝐷𝑦) = (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
8786eleq1d 2811 . . . . . . . 8 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → ((𝐷𝑦) ∈ 𝑥 ↔ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
8887notbid 317 . . . . . . 7 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → (¬ (𝐷𝑦) ∈ 𝑥 ↔ ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
89 fveq2 6890 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝐷𝑧) = (𝐷𝑦))
9089eleq1d 2811 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝐷𝑧) ∈ 𝑥 ↔ (𝐷𝑦) ∈ 𝑥))
9190notbid 317 . . . . . . . 8 (𝑧 = 𝑦 → (¬ (𝐷𝑧) ∈ 𝑥 ↔ ¬ (𝐷𝑦) ∈ 𝑥))
9291cbvrabv 3430 . . . . . . 7 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} = {𝑦 ∈ ω ∣ ¬ (𝐷𝑦) ∈ 𝑥}
9388, 92elrab2 3683 . . . . . 6 ( {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ( {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ ω ∧ ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
9493simprbi 495 . . . . 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 3957 . . 3 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴m 𝑛))) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ (𝐴𝑥))
9728, 96rexlimddv 3151 . 2 ((𝜑𝜓) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ (𝐴𝑥))
9817, 97eqeltrd 2826 1 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  {crab 3419  Vcvv 3462  cdif 3943  wss 3946  c0 4322  ifcif 4523  𝒫 cpw 4597   cint 4946   ciun 4993   class class class wbr 5143   We wwe 5626   × cxp 5670  ccnv 5671  ran crn 5673  Ord word 6364  Oncon0 6365   Fn wfn 6538  wf 6539  1-1wf1 6540  1-1-ontowf1o 6542  cfv 6543  (class class class)co 7413  cmpo 7415  ωcom 7865  m cmap 8844  cdom 8961  csdm 8962  Fincfn 8963  cardccrd 9968
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735  ax-inf2 9674
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-int 4947  df-iun 4995  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6302  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8846  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967
This theorem is referenced by:  pwfseqlem4a  10692  pwfseqlem4  10693
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