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Theorem pwfseqlem3 9672
 Description: Lemma for pwfseq 9676. Using the construction 𝐷 from pwfseqlem1 9670, 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 3341 . . . 4 𝑥 ∈ V
2 vex 3341 . . . 4 𝑟 ∈ V
3 fvex 6360 . . . . 5 (𝐻‘(card‘𝑥)) ∈ V
4 fvex 6360 . . . . 5 (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ V
53, 4ifex 4298 . . . 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 6946 . . . 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 1571 . . 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 483 . . . . . . 7 (𝜓 → ω ≼ 𝑥)
1110adantl 473 . . . . . 6 ((𝜑𝜓) → ω ≼ 𝑥)
12 domnsym 8249 . . . . . 6 (ω ≼ 𝑥 → ¬ 𝑥 ≺ ω)
1311, 12syl 17 . . . . 5 ((𝜑𝜓) → ¬ 𝑥 ≺ ω)
14 isfinite 8720 . . . . 5 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
1513, 14sylnibr 318 . . . 4 ((𝜑𝜓) → ¬ 𝑥 ∈ Fin)
1615iffalsed 4239 . . 3 ((𝜑𝜓) → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) = (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
178, 16syl5eq 2804 . 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 9670 . . . . . 6 ((𝜑𝜓) → 𝐷 ∈ ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∖ 𝑛 ∈ ω (𝑥𝑚 𝑛)))
24 eldif 3723 . . . . . 6 (𝐷 ∈ ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∖ 𝑛 ∈ ω (𝑥𝑚 𝑛)) ↔ (𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)))
2523, 24sylib 208 . . . . 5 ((𝜑𝜓) → (𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛) ∧ ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)))
2625simpld 477 . . . 4 ((𝜑𝜓) → 𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛))
27 eliun 4674 . . . 4 (𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛) ↔ ∃𝑛 ∈ ω 𝐷 ∈ (𝐴𝑚 𝑛))
2826, 27sylib 208 . . 3 ((𝜑𝜓) → ∃𝑛 ∈ ω 𝐷 ∈ (𝐴𝑚 𝑛))
29 elmapi 8043 . . . . . 6 (𝐷 ∈ (𝐴𝑚 𝑛) → 𝐷:𝑛𝐴)
3029ad2antll 767 . . . . 5 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝐷:𝑛𝐴)
31 ssiun2 4713 . . . . . . . . 9 (𝑛 ∈ ω → (𝑥𝑚 𝑛) ⊆ 𝑛 ∈ ω (𝑥𝑚 𝑛))
3231ad2antrl 766 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝑥𝑚 𝑛) ⊆ 𝑛 ∈ ω (𝑥𝑚 𝑛))
3325simprd 482 . . . . . . . . 9 ((𝜑𝜓) → ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛))
3433adantr 472 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛))
3532, 34ssneldd 3745 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ 𝐷 ∈ (𝑥𝑚 𝑛))
36 vex 3341 . . . . . . . . 9 𝑛 ∈ V
371, 36elmap 8050 . . . . . . . 8 (𝐷 ∈ (𝑥𝑚 𝑛) ↔ 𝐷:𝑛𝑥)
38 ffn 6204 . . . . . . . . 9 (𝐷:𝑛𝐴𝐷 Fn 𝑛)
39 ffnfv 6549 . . . . . . . . . 10 (𝐷:𝑛𝑥 ↔ (𝐷 Fn 𝑛 ∧ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
4039baib 982 . . . . . . . . 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 272 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷 ∈ (𝑥𝑚 𝑛) ↔ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
4335, 42mtbid 313 . . . . . 6 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥)
44 nnon 7234 . . . . . . . . 9 (𝑛 ∈ ω → 𝑛 ∈ On)
4544ad2antrl 766 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ On)
46 ssrab2 3826 . . . . . . . . . 10 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ ω
47 omsson 7232 . . . . . . . . . 10 ω ⊆ On
4846, 47sstri 3751 . . . . . . . . 9 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ On
49 ordom 7237 . . . . . . . . . . . . 13 Ord ω
50 simprl 811 . . . . . . . . . . . . 13 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ ω)
51 ordelss 5898 . . . . . . . . . . . . 13 ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
5249, 50, 51sylancr 698 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ⊆ ω)
53 rexnal 3131 . . . . . . . . . . . . 13 (∃𝑧𝑛 ¬ (𝐷𝑧) ∈ 𝑥 ↔ ¬ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥)
5443, 53sylibr 224 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ∃𝑧𝑛 ¬ (𝐷𝑧) ∈ 𝑥)
55 ssrexv 3806 . . . . . . . . . . . 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 4099 . . . . . . . . . . 11 ({𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅ ↔ ∃𝑧 ∈ ω ¬ (𝐷𝑧) ∈ 𝑥)
5856, 57sylibr 224 . . . . . . . . . 10 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅)
59 onint 7158 . . . . . . . . . 10 (({𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ On ∧ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})
6048, 58, 59sylancr 698 . . . . . . . . 9 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})
6148, 60sseldi 3740 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ On)
62 ontri1 5916 . . . . . . . 8 ((𝑛 ∈ On ∧ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ On) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛))
6345, 61, 62syl2anc 696 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛))
64 ssintrab 4650 . . . . . . . 8 (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ∀𝑧 ∈ ω (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧))
65 nnon 7234 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ω → 𝑧 ∈ On)
66 ontri1 5916 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ On ∧ 𝑧 ∈ On) → (𝑛𝑧 ↔ ¬ 𝑧𝑛))
6744, 65, 66syl2an 495 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → (𝑛𝑧 ↔ ¬ 𝑧𝑛))
6867imbi2d 329 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) ↔ (¬ (𝐷𝑧) ∈ 𝑥 → ¬ 𝑧𝑛)))
69 con34b 305 . . . . . . . . . . . . . 14 ((𝑧𝑛 → (𝐷𝑧) ∈ 𝑥) ↔ (¬ (𝐷𝑧) ∈ 𝑥 → ¬ 𝑧𝑛))
7068, 69syl6bbr 278 . . . . . . . . . . . . 13 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) ↔ (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
7170pm5.74da 725 . . . . . . . . . . . 12 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) ↔ (𝑧 ∈ ω → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥))))
72 bi2.04 375 . . . . . . . . . . . 12 ((𝑧 ∈ ω → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)) ↔ (𝑧𝑛 → (𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥)))
7371, 72syl6bb 276 . . . . . . . . . . 11 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) ↔ (𝑧𝑛 → (𝑧 ∈ ω → (𝐷𝑧) ∈ 𝑥))))
74 elnn 7238 . . . . . . . . . . . . . 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 450 . . . . . . . . . . . 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 230 . . . . . . . . . 10 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
8079ad2antrl 766 . . . . . . . . 9 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
8180ralimdv2 3097 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (∀𝑧 ∈ ω (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8264, 81syl5bi 232 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8363, 82sylbird 250 . . . . . 6 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛 → ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
8443, 83mt3d 140 . . . . 5 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛)
8530, 84ffvelrnd 6521 . . . 4 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝐴)
86 fveq2 6350 . . . . . . . . 9 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → (𝐷𝑦) = (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
8786eleq1d 2822 . . . . . . . 8 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → ((𝐷𝑦) ∈ 𝑥 ↔ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
8887notbid 307 . . . . . . 7 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → (¬ (𝐷𝑦) ∈ 𝑥 ↔ ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
89 fveq2 6350 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝐷𝑧) = (𝐷𝑦))
9089eleq1d 2822 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝐷𝑧) ∈ 𝑥 ↔ (𝐷𝑦) ∈ 𝑥))
9190notbid 307 . . . . . . . 8 (𝑧 = 𝑦 → (¬ (𝐷𝑧) ∈ 𝑥 ↔ ¬ (𝐷𝑦) ∈ 𝑥))
9291cbvrabv 3337 . . . . . . 7 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} = {𝑦 ∈ ω ∣ ¬ (𝐷𝑦) ∈ 𝑥}
9388, 92elrab2 3505 . . . . . 6 ( {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ( {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ ω ∧ ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
9493simprbi 483 . . . . 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 3724 . . 3 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ (𝐴𝑥))
9728, 96rexlimddv 3171 . 2 ((𝜑𝜓) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ (𝐴𝑥))
9817, 97eqeltrd 2837 1 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1630   ∈ wcel 2137   ≠ wne 2930  ∀wral 3048  ∃wrex 3049  {crab 3052  Vcvv 3338   ∖ cdif 3710   ⊆ wss 3713  ∅c0 4056  ifcif 4228  𝒫 cpw 4300  ∩ cint 4625  ∪ ciun 4670   class class class wbr 4802   We wwe 5222   × cxp 5262  ◡ccnv 5263  ran crn 5265  Ord word 5881  Oncon0 5882   Fn wfn 6042  ⟶wf 6043  –1-1→wf1 6044  –1-1-onto→wf1o 6046  ‘cfv 6047  (class class class)co 6811   ↦ cmpt2 6813  ωcom 7228   ↑𝑚 cmap 8021   ≼ cdom 8117   ≺ csdm 8118  Fincfn 8119  cardccrd 8949 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-er 7909  df-map 8023  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123 This theorem is referenced by:  pwfseqlem4a  9673  pwfseqlem4  9674
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