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Theorem ingru 9675
Description: The intersection of a universe with a class that acts like a universe is another universe. (Contributed by Mario Carneiro, 10-Jun-2013.)
Assertion
Ref Expression
ingru ((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑈 ∈ Univ → (𝑈𝐴) ∈ Univ))
Distinct variable group:   𝑥,𝑦,𝐴
Allowed substitution hints:   𝑈(𝑥,𝑦)

Proof of Theorem ingru
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 ineq1 3840 . . . . 5 (𝑢 = 𝑈 → (𝑢𝐴) = (𝑈𝐴))
21eleq1d 2715 . . . 4 (𝑢 = 𝑈 → ((𝑢𝐴) ∈ Univ ↔ (𝑈𝐴) ∈ Univ))
32imbi2d 329 . . 3 (𝑢 = 𝑈 → (((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑢𝐴) ∈ Univ) ↔ ((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑈𝐴) ∈ Univ)))
4 elgrug 9652 . . . . . 6 (𝑢 ∈ Univ → (𝑢 ∈ Univ ↔ (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢))))
54ibi 256 . . . . 5 (𝑢 ∈ Univ → (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢)))
6 trin 4796 . . . . . . 7 ((Tr 𝑢 ∧ Tr 𝐴) → Tr (𝑢𝐴))
76ex 449 . . . . . 6 (Tr 𝑢 → (Tr 𝐴 → Tr (𝑢𝐴)))
8 inss1 3866 . . . . . . . 8 (𝑢𝐴) ⊆ 𝑢
9 ssralv 3699 . . . . . . . 8 ((𝑢𝐴) ⊆ 𝑢 → (∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢) → ∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢)))
108, 9ax-mp 5 . . . . . . 7 (∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢) → ∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢))
11 inss2 3867 . . . . . . . 8 (𝑢𝐴) ⊆ 𝐴
12 ssralv 3699 . . . . . . . 8 ((𝑢𝐴) ⊆ 𝐴 → (∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴)) → ∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))))
1311, 12ax-mp 5 . . . . . . 7 (∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴)) → ∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴)))
14 elin 3829 . . . . . . . . . . . . 13 (𝒫 𝑥 ∈ (𝑢𝐴) ↔ (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝐴))
1514simplbi2 654 . . . . . . . . . . . 12 (𝒫 𝑥𝑢 → (𝒫 𝑥𝐴 → 𝒫 𝑥 ∈ (𝑢𝐴)))
16 ssralv 3699 . . . . . . . . . . . . . 14 ((𝑢𝐴) ⊆ 𝑢 → (∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 → ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ 𝑢))
178, 16ax-mp 5 . . . . . . . . . . . . 13 (∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 → ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ 𝑢)
18 ssralv 3699 . . . . . . . . . . . . . 14 ((𝑢𝐴) ⊆ 𝐴 → (∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 → ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ 𝐴))
1911, 18ax-mp 5 . . . . . . . . . . . . 13 (∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 → ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ 𝐴)
20 elin 3829 . . . . . . . . . . . . . . 15 ({𝑥, 𝑦} ∈ (𝑢𝐴) ↔ ({𝑥, 𝑦} ∈ 𝑢 ∧ {𝑥, 𝑦} ∈ 𝐴))
2120simplbi2 654 . . . . . . . . . . . . . 14 ({𝑥, 𝑦} ∈ 𝑢 → ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ∈ (𝑢𝐴)))
2221ral2imi 2976 . . . . . . . . . . . . 13 (∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ 𝑢 → (∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ 𝐴 → ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴)))
2317, 19, 22syl2im 40 . . . . . . . . . . . 12 (∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 → (∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 → ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴)))
2415, 23im2anan9 898 . . . . . . . . . . 11 ((𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢) → ((𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴))))
25 vex 3234 . . . . . . . . . . . . . 14 𝑢 ∈ V
26 mapss 7942 . . . . . . . . . . . . . 14 ((𝑢 ∈ V ∧ (𝑢𝐴) ⊆ 𝑢) → ((𝑢𝐴) ↑𝑚 𝑥) ⊆ (𝑢𝑚 𝑥))
2725, 8, 26mp2an 708 . . . . . . . . . . . . 13 ((𝑢𝐴) ↑𝑚 𝑥) ⊆ (𝑢𝑚 𝑥)
28 ssralv 3699 . . . . . . . . . . . . 13 (((𝑢𝐴) ↑𝑚 𝑥) ⊆ (𝑢𝑚 𝑥) → (∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢 → ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦𝑢))
2927, 28ax-mp 5 . . . . . . . . . . . 12 (∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢 → ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦𝑢)
3025inex1 4832 . . . . . . . . . . . . . . . . 17 (𝑢𝐴) ∈ V
31 vex 3234 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
3230, 31elmap 7928 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ↔ 𝑦:𝑥⟶(𝑢𝐴))
33 fss 6094 . . . . . . . . . . . . . . . . 17 ((𝑦:𝑥⟶(𝑢𝐴) ∧ (𝑢𝐴) ⊆ 𝐴) → 𝑦:𝑥𝐴)
3411, 33mpan2 707 . . . . . . . . . . . . . . . 16 (𝑦:𝑥⟶(𝑢𝐴) → 𝑦:𝑥𝐴)
3532, 34sylbi 207 . . . . . . . . . . . . . . 15 (𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) → 𝑦:𝑥𝐴)
3635imim1i 63 . . . . . . . . . . . . . 14 ((𝑦:𝑥𝐴 ran 𝑦𝐴) → (𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) → ran 𝑦𝐴))
3736alimi 1779 . . . . . . . . . . . . 13 (∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴) → ∀𝑦(𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) → ran 𝑦𝐴))
38 df-ral 2946 . . . . . . . . . . . . 13 (∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦𝐴 ↔ ∀𝑦(𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) → ran 𝑦𝐴))
3937, 38sylibr 224 . . . . . . . . . . . 12 (∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴) → ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦𝐴)
40 elin 3829 . . . . . . . . . . . . . 14 ( ran 𝑦 ∈ (𝑢𝐴) ↔ ( ran 𝑦𝑢 ran 𝑦𝐴))
4140simplbi2 654 . . . . . . . . . . . . 13 ( ran 𝑦𝑢 → ( ran 𝑦𝐴 ran 𝑦 ∈ (𝑢𝐴)))
4241ral2imi 2976 . . . . . . . . . . . 12 (∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦𝑢 → (∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦𝐴 → ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴)))
4329, 39, 42syl2im 40 . . . . . . . . . . 11 (∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢 → (∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴) → ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴)))
4424, 43im2anan9 898 . . . . . . . . . 10 (((𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢) ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢) → (((𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴) ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴)) → ((𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴)) ∧ ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴))))
45443impa 1278 . . . . . . . . 9 ((𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢) → (((𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴) ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴)) → ((𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴)) ∧ ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴))))
46 df-3an 1056 . . . . . . . . 9 ((𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴)) ↔ ((𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴) ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴)))
47 df-3an 1056 . . . . . . . . 9 ((𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴)) ↔ ((𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴)) ∧ ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴)))
4845, 46, 473imtr4g 285 . . . . . . . 8 ((𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢) → ((𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴)) → (𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴))))
4948ral2imi 2976 . . . . . . 7 (∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢) → (∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴)) → ∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴))))
5010, 13, 49syl2im 40 . . . . . 6 (∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢) → (∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴)) → ∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴))))
517, 50im2anan9 898 . . . . 5 ((Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢)) → ((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (Tr (𝑢𝐴) ∧ ∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴)))))
525, 51syl 17 . . . 4 (𝑢 ∈ Univ → ((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (Tr (𝑢𝐴) ∧ ∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴)))))
53 elgrug 9652 . . . . 5 ((𝑢𝐴) ∈ V → ((𝑢𝐴) ∈ Univ ↔ (Tr (𝑢𝐴) ∧ ∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴)))))
5430, 53ax-mp 5 . . . 4 ((𝑢𝐴) ∈ Univ ↔ (Tr (𝑢𝐴) ∧ ∀𝑥 ∈ (𝑢𝐴)(𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦 ∈ (𝑢𝐴))))
5552, 54syl6ibr 242 . . 3 (𝑢 ∈ Univ → ((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑢𝐴) ∈ Univ))
563, 55vtoclga 3303 . 2 (𝑈 ∈ Univ → ((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑈𝐴) ∈ Univ))
5756com12 32 1 ((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑈 ∈ Univ → (𝑈𝐴) ∈ Univ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wal 1521   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191  {cpr 4212   cuni 4468  Tr wtr 4785  ran crn 5144  wf 5922  (class class class)co 6690  𝑚 cmap 7899  Univcgru 9650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-gru 9651
This theorem is referenced by:  wfgru  9676
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