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Theorem pwfseqlem3 9426
Description: Lemma for pwfseq 9430. Using the construction 𝐷 from pwfseqlem1 9424, 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 3189 . . . 4 𝑥 ∈ V
2 vex 3189 . . . 4 𝑟 ∈ V
3 fvex 6158 . . . . 5 (𝐻‘(card‘𝑥)) ∈ V
4 fvex 6158 . . . . 5 (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ V
53, 4ifex 4128 . . . 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 6736 . . . 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 1421 . . 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 480 . . . . . . 7 (𝜓 → ω ≼ 𝑥)
1110adantl 482 . . . . . 6 ((𝜑𝜓) → ω ≼ 𝑥)
12 domnsym 8030 . . . . . 6 (ω ≼ 𝑥 → ¬ 𝑥 ≺ ω)
1311, 12syl 17 . . . . 5 ((𝜑𝜓) → ¬ 𝑥 ≺ ω)
14 isfinite 8493 . . . . 5 (𝑥 ∈ Fin ↔ 𝑥 ≺ ω)
1513, 14sylnibr 319 . . . 4 ((𝜑𝜓) → ¬ 𝑥 ∈ Fin)
1615iffalsed 4069 . . 3 ((𝜑𝜓) → if(𝑥 ∈ Fin, (𝐻‘(card‘𝑥)), (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})) = (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
178, 16syl5eq 2667 . 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 9424 . . . . . 6 ((𝜑𝜓) → 𝐷 ∈ ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∖ 𝑛 ∈ ω (𝑥𝑚 𝑛)))
24 eldif 3565 . . . . . 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 475 . . . 4 ((𝜑𝜓) → 𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛))
27 eliun 4490 . . . 4 (𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛) ↔ ∃𝑛 ∈ ω 𝐷 ∈ (𝐴𝑚 𝑛))
2826, 27sylib 208 . . 3 ((𝜑𝜓) → ∃𝑛 ∈ ω 𝐷 ∈ (𝐴𝑚 𝑛))
29 elmapi 7823 . . . . . 6 (𝐷 ∈ (𝐴𝑚 𝑛) → 𝐷:𝑛𝐴)
3029ad2antll 764 . . . . 5 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝐷:𝑛𝐴)
31 ssiun2 4529 . . . . . . . . 9 (𝑛 ∈ ω → (𝑥𝑚 𝑛) ⊆ 𝑛 ∈ ω (𝑥𝑚 𝑛))
3231ad2antrl 763 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝑥𝑚 𝑛) ⊆ 𝑛 ∈ ω (𝑥𝑚 𝑛))
3325simprd 479 . . . . . . . . 9 ((𝜑𝜓) → ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛))
3433adantr 481 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛))
3532, 34ssneldd 3586 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ 𝐷 ∈ (𝑥𝑚 𝑛))
36 vex 3189 . . . . . . . . 9 𝑛 ∈ V
371, 36elmap 7830 . . . . . . . 8 (𝐷 ∈ (𝑥𝑚 𝑛) ↔ 𝐷:𝑛𝑥)
38 ffn 6002 . . . . . . . . 9 (𝐷:𝑛𝐴𝐷 Fn 𝑛)
39 ffnfv 6343 . . . . . . . . . 10 (𝐷:𝑛𝑥 ↔ (𝐷 Fn 𝑛 ∧ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥))
4039baib 943 . . . . . . . . 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 314 . . . . . 6 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ¬ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥)
44 nnon 7018 . . . . . . . . 9 (𝑛 ∈ ω → 𝑛 ∈ On)
4544ad2antrl 763 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ On)
46 ssrab2 3666 . . . . . . . . . 10 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ ω
47 omsson 7016 . . . . . . . . . 10 ω ⊆ On
4846, 47sstri 3592 . . . . . . . . 9 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ On
49 ordom 7021 . . . . . . . . . . . . 13 Ord ω
50 simprl 793 . . . . . . . . . . . . 13 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ∈ ω)
51 ordelss 5698 . . . . . . . . . . . . 13 ((Ord ω ∧ 𝑛 ∈ ω) → 𝑛 ⊆ ω)
5249, 50, 51sylancr 694 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → 𝑛 ⊆ ω)
53 rexnal 2989 . . . . . . . . . . . . 13 (∃𝑧𝑛 ¬ (𝐷𝑧) ∈ 𝑥 ↔ ¬ ∀𝑧𝑛 (𝐷𝑧) ∈ 𝑥)
5443, 53sylibr 224 . . . . . . . . . . . 12 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ∃𝑧𝑛 ¬ (𝐷𝑧) ∈ 𝑥)
55 ssrexv 3646 . . . . . . . . . . . 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 3932 . . . . . . . . . . 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 6942 . . . . . . . . . 10 (({𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ⊆ On ∧ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ≠ ∅) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})
6048, 58, 59sylancr 694 . . . . . . . . 9 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥})
6148, 60sseldi 3581 . . . . . . . 8 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ On)
62 ontri1 5716 . . . . . . . 8 ((𝑛 ∈ On ∧ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ On) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛))
6345, 61, 62syl2anc 692 . . . . . . 7 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ¬ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ 𝑛))
64 ssintrab 4465 . . . . . . . 8 (𝑛 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ∀𝑧 ∈ ω (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧))
65 nnon 7018 . . . . . . . . . . . . . . . 16 (𝑧 ∈ ω → 𝑧 ∈ On)
66 ontri1 5716 . . . . . . . . . . . . . . . 16 ((𝑛 ∈ On ∧ 𝑧 ∈ On) → (𝑛𝑧 ↔ ¬ 𝑧𝑛))
6744, 65, 66syl2an 494 . . . . . . . . . . . . . . 15 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → (𝑛𝑧 ↔ ¬ 𝑧𝑛))
6867imbi2d 330 . . . . . . . . . . . . . 14 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) ↔ (¬ (𝐷𝑧) ∈ 𝑥 → ¬ 𝑧𝑛)))
69 con34b 306 . . . . . . . . . . . . . 14 ((𝑧𝑛 → (𝐷𝑧) ∈ 𝑥) ↔ (¬ (𝐷𝑧) ∈ 𝑥 → ¬ 𝑧𝑛))
7068, 69syl6bbr 278 . . . . . . . . . . . . 13 ((𝑛 ∈ ω ∧ 𝑧 ∈ ω) → ((¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧) ↔ (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
7170pm5.74da 722 . . . . . . . . . . . 12 (𝑛 ∈ ω → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) ↔ (𝑧 ∈ ω → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥))))
72 bi2.04 376 . . . . . . . . . . . 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 7022 . . . . . . . . . . . . . 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 451 . . . . . . . . . . . 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 763 . . . . . . . . 9 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → ((𝑧 ∈ ω → (¬ (𝐷𝑧) ∈ 𝑥𝑛𝑧)) → (𝑧𝑛 → (𝐷𝑧) ∈ 𝑥)))
8180ralimdv2 2955 . . . . . . . 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 6316 . . . 4 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝐴)
86 fveq2 6148 . . . . . . . . 9 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → (𝐷𝑦) = (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}))
8786eleq1d 2683 . . . . . . . 8 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → ((𝐷𝑦) ∈ 𝑥 ↔ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
8887notbid 308 . . . . . . 7 (𝑦 = {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} → (¬ (𝐷𝑦) ∈ 𝑥 ↔ ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
89 fveq2 6148 . . . . . . . . . 10 (𝑧 = 𝑦 → (𝐷𝑧) = (𝐷𝑦))
9089eleq1d 2683 . . . . . . . . 9 (𝑧 = 𝑦 → ((𝐷𝑧) ∈ 𝑥 ↔ (𝐷𝑦) ∈ 𝑥))
9190notbid 308 . . . . . . . 8 (𝑧 = 𝑦 → (¬ (𝐷𝑧) ∈ 𝑥 ↔ ¬ (𝐷𝑦) ∈ 𝑥))
9291cbvrabv 3185 . . . . . . 7 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} = {𝑦 ∈ ω ∣ ¬ (𝐷𝑦) ∈ 𝑥}
9388, 92elrab2 3348 . . . . . 6 ( {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ↔ ( {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥} ∈ ω ∧ ¬ (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ 𝑥))
9493simprbi 480 . . . . 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 3566 . . 3 (((𝜑𝜓) ∧ (𝑛 ∈ ω ∧ 𝐷 ∈ (𝐴𝑚 𝑛))) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ (𝐴𝑥))
9728, 96rexlimddv 3028 . 2 ((𝜑𝜓) → (𝐷 {𝑧 ∈ ω ∣ ¬ (𝐷𝑧) ∈ 𝑥}) ∈ (𝐴𝑥))
9817, 97eqeltrd 2698 1 ((𝜑𝜓) → (𝑥𝐹𝑟) ∈ (𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3186  cdif 3552  wss 3555  c0 3891  ifcif 4058  𝒫 cpw 4130   cint 4440   ciun 4485   class class class wbr 4613   We wwe 5032   × cxp 5072  ccnv 5073  ran crn 5075  Ord word 5681  Oncon0 5682   Fn wfn 5842  wf 5843  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  cmpt2 6606  ωcom 7012  𝑚 cmap 7802  cdom 7897  csdm 7898  Fincfn 7899  cardccrd 8705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903
This theorem is referenced by:  pwfseqlem4a  9427  pwfseqlem4  9428
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