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Theorem pwfseqlem1 9768
Description: Lemma for pwfseq 9774. 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 473 . . . . 5 ((𝜑𝜓) → 𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛))
4 f1f 6316 . . . . 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 3883 . . . . . 6 {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ⊆ 𝑥
7 pwfseqlem4.ps . . . . . . 7 (𝜓 ↔ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥))
8 simprl1 1282 . . . . . . 7 ((𝜑 ∧ ((𝑥𝐴𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥) ∧ ω ≼ 𝑥)) → 𝑥𝐴)
97, 8sylan2b 588 . . . . . 6 ((𝜑𝜓) → 𝑥𝐴)
106, 9syl5ss 3809 . . . . 5 ((𝜑𝜓) → {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ⊆ 𝐴)
11 vex 3388 . . . . . . 7 𝑥 ∈ V
1211rabex 5007 . . . . . 6 {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ∈ V
1312elpw 4355 . . . . 5 ({𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ∈ 𝒫 𝐴 ↔ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ⊆ 𝐴)
1410, 13sylibr 226 . . . 4 ((𝜑𝜓) → {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ∈ 𝒫 𝐴)
155, 14ffvelrnd 6586 . . 3 ((𝜑𝜓) → (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}) ∈ 𝑛 ∈ ω (𝐴𝑚 𝑛))
161, 15syl5eqel 2882 . 2 ((𝜑𝜓) → 𝐷 𝑛 ∈ ω (𝐴𝑚 𝑛))
17 pm5.19 377 . . 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 473 . . . . . . . 8 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → 𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1𝑥)
20 f1f 6316 . . . . . . . 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 6583 . . . . . . 7 ((𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)⟶𝑥𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾𝐷) ∈ 𝑥)
2321, 22sylancom 583 . . . . . 6 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾𝐷) ∈ 𝑥)
24 f1f1orn 6367 . . . . . . . . 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 6760 . . . . . . . 8 ((𝐾: 𝑛 ∈ ω (𝑥𝑚 𝑛)–1-1-onto→ran 𝐾𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) = 𝐷)
2725, 26sylancom 583 . . . . . . 7 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) = 𝐷)
28 f1fn 6317 . . . . . . . . . . 11 (𝐺:𝒫 𝐴1-1 𝑛 ∈ ω (𝐴𝑚 𝑛) → 𝐺 Fn 𝒫 𝐴)
293, 28syl 17 . . . . . . . . . 10 ((𝜑𝜓) → 𝐺 Fn 𝒫 𝐴)
30 fnfvelrn 6582 . . . . . . . . . 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 580 . . . . . . . . 9 ((𝜑𝜓) → (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}) ∈ ran 𝐺)
321, 31syl5eqel 2882 . . . . . . . 8 ((𝜑𝜓) → 𝐷 ∈ ran 𝐺)
3332adantr 473 . . . . . . 7 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → 𝐷 ∈ ran 𝐺)
3427, 33eqeltrd 2878 . . . . . 6 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) ∈ ran 𝐺)
35 fveq2 6411 . . . . . . . . . . 11 (𝑦 = (𝐾𝐷) → (𝐾𝑦) = (𝐾‘(𝐾𝐷)))
3635eleq1d 2863 . . . . . . . . . 10 (𝑦 = (𝐾𝐷) → ((𝐾𝑦) ∈ ran 𝐺 ↔ (𝐾‘(𝐾𝐷)) ∈ ran 𝐺))
37 id 22 . . . . . . . . . . . 12 (𝑦 = (𝐾𝐷) → 𝑦 = (𝐾𝐷))
38 2fveq3 6416 . . . . . . . . . . . 12 (𝑦 = (𝐾𝐷) → (𝐺‘(𝐾𝑦)) = (𝐺‘(𝐾‘(𝐾𝐷))))
3937, 38eleq12d 2872 . . . . . . . . . . 11 (𝑦 = (𝐾𝐷) → (𝑦 ∈ (𝐺‘(𝐾𝑦)) ↔ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
4039notbid 310 . . . . . . . . . 10 (𝑦 = (𝐾𝐷) → (¬ 𝑦 ∈ (𝐺‘(𝐾𝑦)) ↔ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
4136, 40anbi12d 625 . . . . . . . . 9 (𝑦 = (𝐾𝐷) → (((𝐾𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦))) ↔ ((𝐾‘(𝐾𝐷)) ∈ ran 𝐺 ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))))))
42 fveq2 6411 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝐾𝑤) = (𝐾𝑦))
4342eleq1d 2863 . . . . . . . . . . 11 (𝑤 = 𝑦 → ((𝐾𝑤) ∈ ran 𝐺 ↔ (𝐾𝑦) ∈ ran 𝐺))
44 id 22 . . . . . . . . . . . . 13 (𝑤 = 𝑦𝑤 = 𝑦)
45 2fveq3 6416 . . . . . . . . . . . . 13 (𝑤 = 𝑦 → (𝐺‘(𝐾𝑤)) = (𝐺‘(𝐾𝑦)))
4644, 45eleq12d 2872 . . . . . . . . . . . 12 (𝑤 = 𝑦 → (𝑤 ∈ (𝐺‘(𝐾𝑤)) ↔ 𝑦 ∈ (𝐺‘(𝐾𝑦))))
4746notbid 310 . . . . . . . . . . 11 (𝑤 = 𝑦 → (¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)) ↔ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦))))
4843, 47anbi12d 625 . . . . . . . . . 10 (𝑤 = 𝑦 → (((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤))) ↔ ((𝐾𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦)))))
4948cbvrabv 3383 . . . . . . . . 9 {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} = {𝑦𝑥 ∣ ((𝐾𝑦) ∈ ran 𝐺 ∧ ¬ 𝑦 ∈ (𝐺‘(𝐾𝑦)))}
5041, 49elrab2 3560 . . . . . . . 8 ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ((𝐾𝐷) ∈ 𝑥 ∧ ((𝐾‘(𝐾𝐷)) ∈ ran 𝐺 ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))))))
51 anass 461 . . . . . . . 8 ((((𝐾𝐷) ∈ 𝑥 ∧ (𝐾‘(𝐾𝐷)) ∈ ran 𝐺) ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))) ↔ ((𝐾𝐷) ∈ 𝑥 ∧ ((𝐾‘(𝐾𝐷)) ∈ ran 𝐺 ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))))))
5250, 51bitr4i 270 . . . . . . 7 ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ (((𝐾𝐷) ∈ 𝑥 ∧ (𝐾‘(𝐾𝐷)) ∈ ran 𝐺) ∧ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
5352baib 532 . . . . . 6 (((𝐾𝐷) ∈ 𝑥 ∧ (𝐾‘(𝐾𝐷)) ∈ ran 𝐺) → ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
5423, 34, 53syl2anc 580 . . . . 5 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷)))))
5527, 1syl6eq 2849 . . . . . . . . 9 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐾‘(𝐾𝐷)) = (𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}))
5655fveq2d 6415 . . . . . . . 8 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐺‘(𝐾‘(𝐾𝐷))) = (𝐺‘(𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})))
57 f1f1orn 6367 . . . . . . . . . . 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 6760 . . . . . . . . . 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 580 . . . . . . . . 9 ((𝜑𝜓) → (𝐺‘(𝐺‘{𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})) = {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
6160adantr 473 . . . . . . . 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 2833 . . . . . . 7 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (𝐺‘(𝐾‘(𝐾𝐷))) = {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))})
6362eleq2d 2864 . . . . . 6 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → ((𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))) ↔ (𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}))
6463notbid 310 . . . . 5 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → (¬ (𝐾𝐷) ∈ (𝐺‘(𝐾‘(𝐾𝐷))) ↔ ¬ (𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}))
6554, 64bitrd 271 . . . 4 (((𝜑𝜓) ∧ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛)) → ((𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))} ↔ ¬ (𝐾𝐷) ∈ {𝑤𝑥 ∣ ((𝐾𝑤) ∈ ran 𝐺 ∧ ¬ 𝑤 ∈ (𝐺‘(𝐾𝑤)))}))
6665ex 402 . . 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 191 . 2 ((𝜑𝜓) → ¬ 𝐷 𝑛 ∈ ω (𝑥𝑚 𝑛))
6816, 67eldifd 3780 1 ((𝜑𝜓) → 𝐷 ∈ ( 𝑛 ∈ ω (𝐴𝑚 𝑛) ∖ 𝑛 ∈ ω (𝑥𝑚 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  {crab 3093  cdif 3766  wss 3769  𝒫 cpw 4349   ciun 4710   class class class wbr 4843   We wwe 5270   × cxp 5310  ccnv 5311  ran crn 5313   Fn wfn 6096  wf 6097  1-1wf1 6098  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6878  ωcom 7299  𝑚 cmap 8095  cdom 8193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109
This theorem is referenced by:  pwfseqlem3  9770
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