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Theorem pwfseqlem1 9682
Description: Lemma for pwfseq 9688. Derive a contradiction by diagonalization. (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)))})
Assertion
Ref Expression
pwfseqlem1 ((𝜑𝜓) → 𝐷 ∈ ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∖ 𝑛 ∈ ω (𝑥𝑚 𝑛)))
Distinct variable groups:   𝑛,𝑟,𝑤,𝑥   𝐷,𝑛   𝑤,𝐺   𝑤,𝐾   𝐻,𝑟,𝑥   𝜑,𝑛,𝑟,𝑥   𝜓,𝑛   𝐴,𝑛,𝑟,𝑥
Allowed substitution hints:   𝜑(𝑤)   𝜓(𝑥,𝑤,𝑟)   𝐴(𝑤)   𝐷(𝑥,𝑤,𝑟)   𝐺(𝑥,𝑛,𝑟)   𝐻(𝑤,𝑛)   𝐾(𝑥,𝑛,𝑟)   𝑋(𝑥,𝑤,𝑛,𝑟)

Proof of Theorem pwfseqlem1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 pwfseqlem4.d . . 3 𝐷 = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
2 pwfseqlem4.g . . . . . 6 (𝜑𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
32adantr 466 . . . . 5 ((𝜑𝜓) → 𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
4 f1f 6241 . . . . 5 (𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) → 𝐺:𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛))
53, 4syl 17 . . . 4 ((𝜑𝜓) → 𝐺:𝒫 𝐴 𝑛 ∈ ω (𝐴𝑚 𝑛))
6 ssrab2 3836 . . . . . 6 {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ⊆ 𝑥
7 pwfseqlem4.ps . . . . . . 7 (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
8 simprl1 1266 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) → 𝑥𝐴)
97, 8sylan2b 581 . . . . . 6 ((𝜑𝜓) → 𝑥𝐴)
106, 9syl5ss 3763 . . . . 5 ((𝜑𝜓) → {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ⊆ 𝐴)
11 vex 3354 . . . . . . 7 𝑥 ∈ V
1211rabex 4946 . . . . . 6 {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ∈ V
1312elpw 4303 . . . . 5 ({𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ∈ 𝒫 𝐴 ↔ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ⊆ 𝐴)
1410, 13sylibr 224 . . . 4 ((𝜑𝜓) → {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ∈ 𝒫 𝐴)
155, 14ffvelrnd 6503 . . 3 ((𝜑𝜓) → (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}) ∈ 𝑛 ∈ ω (𝐴𝑚 𝑛))
161, 15syl5eqel 2854 . 2 ((𝜑𝜓) → 𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛))
17 pm5.19 374 . . 3 ¬ ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ¬ (𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
18 pwfseqlem4.k . . . . . . . . 9 ((𝜑𝜓) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)
1918adantr 466 . . . . . . . 8 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)
20 f1f 6241 . . . . . . . 8 (𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)⟶𝑥)
2119, 20syl 17 . . . . . . 7 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)⟶𝑥)
22 ffvelrn 6500 . . . . . . 7 ((𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)⟶𝑥𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾𝐷) ∈ 𝑥)
2321, 22sylancom 576 . . . . . 6 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾𝐷) ∈ 𝑥)
24 f1f1orn 6289 . . . . . . . . 9 (𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1-onto→ran 𝐾)
2519, 24syl 17 . . . . . . . 8 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1-onto→ran 𝐾)
26 f1ocnvfv1 6675 . . . . . . . 8 ((𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1-onto→ran 𝐾𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) = 𝐷)
2725, 26sylancom 576 . . . . . . 7 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) = 𝐷)
28 f1fn 6242 . . . . . . . . . . 11 (𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) → 𝐺 Fn 𝒫 𝐴)
293, 28syl 17 . . . . . . . . . 10 ((𝜑𝜓) → 𝐺 Fn 𝒫 𝐴)
30 fnfvelrn 6499 . . . . . . . . . 10 ((𝐺 Fn 𝒫 𝐴 ∧ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ∈ 𝒫 𝐴) → (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}) ∈ ran 𝐺)
3129, 14, 30syl2anc 573 . . . . . . . . 9 ((𝜑𝜓) → (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}) ∈ ran 𝐺)
321, 31syl5eqel 2854 . . . . . . . 8 ((𝜑𝜓) → 𝐷 ∈ ran 𝐺)
3332adantr 466 . . . . . . 7 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → 𝐷 ∈ ran 𝐺)
3427, 33eqeltrd 2850 . . . . . 6 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) ∈ ran 𝐺)
35 fveq2 6332 . . . . . . . . . . 11 (𝑦 = (𝐾𝐷) → (𝐾𝑦) = (𝐾‘(𝐾𝐷)))
3635eleq1d 2835 . . . . . . . . . 10 (𝑦 = (𝐾𝐷) → ((𝐾𝑦) ∈ ran 𝐺 ↔ (𝐾‘(𝐾𝐷)) ∈ ran 𝐺))
37 id 22 . . . . . . . . . . . 12 (𝑦 = (𝐾𝐷) → 𝑦 = (𝐾𝐷))
3835fveq2d 6336 . . . . . . . . . . . 12 (𝑦 = (𝐾𝐷) → (𝐺‘(𝐾𝑦)) = (𝐺‘(𝐾‘(𝐾𝐷))))
3937, 38eleq12d 2844 . . . . . . . . . . 11 (𝑦 = (𝐾𝐷) → (𝑦 ∈ (𝐺‘(𝐾𝑦)) ↔ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
4039notbid 307 . . . . . . . . . 10 (𝑦 = (𝐾𝐷) → (¬ 𝑦 ∈ (𝐺‘(𝐾𝑦)) ↔ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
4136, 40anbi12d 616 . . . . . . . . 9 (𝑦 = (𝐾𝐷) → (((𝐾𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦))) ↔ ((𝐾‘(𝐾𝐷)) ∈ ran 𝐺 ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))))))
42 fveq2 6332 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝐾𝑤) = (𝐾𝑦))
4342eleq1d 2835 . . . . . . . . . . 11 (𝑤 = 𝑦 → ((𝐾𝑤) ∈ ran 𝐺 ↔ (𝐾𝑦) ∈ ran 𝐺))
44 id 22 . . . . . . . . . . . . 13 (𝑤 = 𝑦𝑤 = 𝑦)
4542fveq2d 6336 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (𝐺‘(𝐾𝑤)) = (𝐺‘(𝐾𝑦)))
4644, 45eleq12d 2844 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝑤 ∈ (𝐺‘(𝐾𝑤)) ↔ 𝑦 ∈ (𝐺‘(𝐾𝑦))))
4746notbid 307 . . . . . . . . . . 11 (𝑤 = 𝑦 → (¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)) ↔ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦))))
4843, 47anbi12d 616 . . . . . . . . . 10 (𝑤 = 𝑦 → (((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤))) ↔ ((𝐾𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦)))))
4948cbvrabv 3349 . . . . . . . . 9 {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} = {𝑦𝑥 ∣ ((𝐾𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦)))}
5041, 49elrab2 3518 . . . . . . . 8 ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ((𝐾𝐷) ∈ 𝑥 ∧ ((𝐾‘(𝐾𝐷)) ∈ ran 𝐺 ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))))))
51 anass 459 . . . . . . . 8 ((((𝐾𝐷) ∈ 𝑥 ∧ (𝐾‘(𝐾𝐷)) ∈ ran 𝐺) ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))) ↔ ((𝐾𝐷) ∈ 𝑥 ∧ ((𝐾‘(𝐾𝐷)) ∈ ran 𝐺 ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))))))
5250, 51bitr4i 267 . . . . . . 7 ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ (((𝐾𝐷) ∈ 𝑥 ∧ (𝐾‘(𝐾𝐷)) ∈ ran 𝐺) ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
5352baib 525 . . . . . 6 (((𝐾𝐷) ∈ 𝑥 ∧ (𝐾‘(𝐾𝐷)) ∈ ran 𝐺) → ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
5423, 34, 53syl2anc 573 . . . . 5 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
5527, 1syl6eq 2821 . . . . . . . . 9 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}))
5655fveq2d 6336 . . . . . . . 8 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐺‘(𝐾‘(𝐾𝐷))) = (𝐺‘(𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})))
57 f1f1orn 6289 . . . . . . . . . . 11 (𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) → 𝐺:𝒫 𝐴1-1-onto→ran 𝐺)
583, 57syl 17 . . . . . . . . . 10 ((𝜑𝜓) → 𝐺:𝒫 𝐴1-1-onto→ran 𝐺)
59 f1ocnvfv1 6675 . . . . . . . . . 10 ((𝐺:𝒫 𝐴1-1-onto→ran 𝐺 ∧ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ∈ 𝒫 𝐴) → (𝐺‘(𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})) = {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
6058, 14, 59syl2anc 573 . . . . . . . . 9 ((𝜑𝜓) → (𝐺‘(𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})) = {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
6160adantr 466 . . . . . . . 8 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐺‘(𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})) = {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
6256, 61eqtrd 2805 . . . . . . 7 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐺‘(𝐾‘(𝐾𝐷))) = {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
6362eleq2d 2836 . . . . . 6 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → ((𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))) ↔ (𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}))
6463notbid 307 . . . . 5 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))) ↔ ¬ (𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}))
6554, 64bitrd 268 . . . 4 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ¬ (𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}))
6665ex 397 . . 3 ((𝜑𝜓) → (𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛) → ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ¬ (𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})))
6717, 66mtoi 190 . 2 ((𝜑𝜓) → ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛))
6816, 67eldifd 3734 1 ((𝜑𝜓) → 𝐷 ∈ ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∖ 𝑛 ∈ ω (𝑥𝑚 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  {crab 3065  cdif 3720  wss 3723  𝒫 cpw 4297   ciun 4654   class class class wbr 4786   We wwe 5207   × cxp 5247  ccnv 5248  ran crn 5250   Fn wfn 6026  wf 6027  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6793  ωcom 7212  𝑚 cmap 8009  cdom 8107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039
This theorem is referenced by:  pwfseqlem3  9684
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