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Theorem pwfseqlem1 9424
Description: Lemma for pwfseq 9430. 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 481 . . . . 5 ((𝜑𝜓) → 𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
4 f1f 6058 . . . . 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 3666 . . . . . 6 {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ⊆ 𝑥
7 pwfseqlem4.ps . . . . . . 7 (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
8 simprl1 1104 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) → 𝑥𝐴)
97, 8sylan2b 492 . . . . . 6 ((𝜑𝜓) → 𝑥𝐴)
106, 9syl5ss 3594 . . . . 5 ((𝜑𝜓) → {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ⊆ 𝐴)
11 vex 3189 . . . . . . 7 𝑥 ∈ V
1211rabex 4773 . . . . . 6 {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ∈ V
1312elpw 4136 . . . . 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 6316 . . 3 ((𝜑𝜓) → (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}) ∈ 𝑛 ∈ ω (𝐴𝑚 𝑛))
161, 15syl5eqel 2702 . 2 ((𝜑𝜓) → 𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛))
17 pm5.19 375 . . 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 481 . . . . . . . 8 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)
20 f1f 6058 . . . . . . . 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 6313 . . . . . . 7 ((𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)⟶𝑥𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾𝐷) ∈ 𝑥)
2321, 22sylancom 700 . . . . . 6 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾𝐷) ∈ 𝑥)
24 f1f1orn 6105 . . . . . . . . 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 6486 . . . . . . . 8 ((𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1-onto→ran 𝐾𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) = 𝐷)
2725, 26sylancom 700 . . . . . . 7 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) = 𝐷)
28 f1fn 6059 . . . . . . . . . . 11 (𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) → 𝐺 Fn 𝒫 𝐴)
293, 28syl 17 . . . . . . . . . 10 ((𝜑𝜓) → 𝐺 Fn 𝒫 𝐴)
30 fnfvelrn 6312 . . . . . . . . . 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 692 . . . . . . . . 9 ((𝜑𝜓) → (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}) ∈ ran 𝐺)
321, 31syl5eqel 2702 . . . . . . . 8 ((𝜑𝜓) → 𝐷 ∈ ran 𝐺)
3332adantr 481 . . . . . . 7 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → 𝐷 ∈ ran 𝐺)
3427, 33eqeltrd 2698 . . . . . 6 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) ∈ ran 𝐺)
35 fveq2 6148 . . . . . . . . . . 11 (𝑦 = (𝐾𝐷) → (𝐾𝑦) = (𝐾‘(𝐾𝐷)))
3635eleq1d 2683 . . . . . . . . . 10 (𝑦 = (𝐾𝐷) → ((𝐾𝑦) ∈ ran 𝐺 ↔ (𝐾‘(𝐾𝐷)) ∈ ran 𝐺))
37 id 22 . . . . . . . . . . . 12 (𝑦 = (𝐾𝐷) → 𝑦 = (𝐾𝐷))
3835fveq2d 6152 . . . . . . . . . . . 12 (𝑦 = (𝐾𝐷) → (𝐺‘(𝐾𝑦)) = (𝐺‘(𝐾‘(𝐾𝐷))))
3937, 38eleq12d 2692 . . . . . . . . . . 11 (𝑦 = (𝐾𝐷) → (𝑦 ∈ (𝐺‘(𝐾𝑦)) ↔ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
4039notbid 308 . . . . . . . . . 10 (𝑦 = (𝐾𝐷) → (¬ 𝑦 ∈ (𝐺‘(𝐾𝑦)) ↔ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
4136, 40anbi12d 746 . . . . . . . . 9 (𝑦 = (𝐾𝐷) → (((𝐾𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦))) ↔ ((𝐾‘(𝐾𝐷)) ∈ ran 𝐺 ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))))))
42 fveq2 6148 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝐾𝑤) = (𝐾𝑦))
4342eleq1d 2683 . . . . . . . . . . 11 (𝑤 = 𝑦 → ((𝐾𝑤) ∈ ran 𝐺 ↔ (𝐾𝑦) ∈ ran 𝐺))
44 id 22 . . . . . . . . . . . . 13 (𝑤 = 𝑦𝑤 = 𝑦)
4542fveq2d 6152 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (𝐺‘(𝐾𝑤)) = (𝐺‘(𝐾𝑦)))
4644, 45eleq12d 2692 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝑤 ∈ (𝐺‘(𝐾𝑤)) ↔ 𝑦 ∈ (𝐺‘(𝐾𝑦))))
4746notbid 308 . . . . . . . . . . 11 (𝑤 = 𝑦 → (¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)) ↔ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦))))
4843, 47anbi12d 746 . . . . . . . . . 10 (𝑤 = 𝑦 → (((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤))) ↔ ((𝐾𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦)))))
4948cbvrabv 3185 . . . . . . . . 9 {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} = {𝑦𝑥 ∣ ((𝐾𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦)))}
5041, 49elrab2 3348 . . . . . . . 8 ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ((𝐾𝐷) ∈ 𝑥 ∧ ((𝐾‘(𝐾𝐷)) ∈ ran 𝐺 ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))))))
51 anass 680 . . . . . . . 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 943 . . . . . 6 (((𝐾𝐷) ∈ 𝑥 ∧ (𝐾‘(𝐾𝐷)) ∈ ran 𝐺) → ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
5423, 34, 53syl2anc 692 . . . . 5 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
5527, 1syl6eq 2671 . . . . . . . . 9 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}))
5655fveq2d 6152 . . . . . . . 8 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐺‘(𝐾‘(𝐾𝐷))) = (𝐺‘(𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})))
57 f1f1orn 6105 . . . . . . . . . . 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 6486 . . . . . . . . . 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 692 . . . . . . . . 9 ((𝜑𝜓) → (𝐺‘(𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})) = {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
6160adantr 481 . . . . . . . 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 2655 . . . . . . 7 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐺‘(𝐾‘(𝐾𝐷))) = {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
6362eleq2d 2684 . . . . . 6 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → ((𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))) ↔ (𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}))
6463notbid 308 . . . . 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 450 . . 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 3566 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 384  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  cdif 3552  wss 3555  𝒫 cpw 4130   ciun 4485   class class class wbr 4613   We wwe 5032   × cxp 5072  ccnv 5073  ran crn 5075   Fn wfn 5842  wf 5843  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  ωcom 7012  𝑚 cmap 7802  cdom 7897
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
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  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-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855
This theorem is referenced by:  pwfseqlem3  9426
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