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Theorem ingru 10035
 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 4068 . . . . 5 (𝑢 = 𝑈 → (𝑢𝐴) = (𝑈𝐴))
21eleq1d 2850 . . . 4 (𝑢 = 𝑈 → ((𝑢𝐴) ∈ Univ ↔ (𝑈𝐴) ∈ Univ))
32imbi2d 333 . . 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 10012 . . . . . 6 (𝑢 ∈ Univ → (𝑢 ∈ Univ ↔ (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢))))
54ibi 259 . . . . 5 (𝑢 ∈ Univ → (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢𝑚 𝑥) ran 𝑦𝑢)))
6 trin 5040 . . . . . . 7 ((Tr 𝑢 ∧ Tr 𝐴) → Tr (𝑢𝐴))
76ex 405 . . . . . 6 (Tr 𝑢 → (Tr 𝐴 → Tr (𝑢𝐴)))
8 inss1 4092 . . . . . . . 8 (𝑢𝐴) ⊆ 𝑢
9 ssralv 3923 . . . . . . . 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 4093 . . . . . . . 8 (𝑢𝐴) ⊆ 𝐴
12 ssralv 3923 . . . . . . . 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 4057 . . . . . . . . . . . . 13 (𝒫 𝑥 ∈ (𝑢𝐴) ↔ (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝐴))
1514simplbi2 493 . . . . . . . . . . . 12 (𝒫 𝑥𝑢 → (𝒫 𝑥𝐴 → 𝒫 𝑥 ∈ (𝑢𝐴)))
16 ssralv 3923 . . . . . . . . . . . . . 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 3923 . . . . . . . . . . . . . 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 4057 . . . . . . . . . . . . . . 15 ({𝑥, 𝑦} ∈ (𝑢𝐴) ↔ ({𝑥, 𝑦} ∈ 𝑢 ∧ {𝑥, 𝑦} ∈ 𝐴))
2120simplbi2 493 . . . . . . . . . . . . . 14 ({𝑥, 𝑦} ∈ 𝑢 → ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ∈ (𝑢𝐴)))
2221ral2imi 3106 . . . . . . . . . . . . 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 610 . . . . . . . . . . 11 ((𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢) → ((𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴))))
25 vex 3418 . . . . . . . . . . . . . 14 𝑢 ∈ V
26 mapss 8251 . . . . . . . . . . . . . 14 ((𝑢 ∈ V ∧ (𝑢𝐴) ⊆ 𝑢) → ((𝑢𝐴) ↑𝑚 𝑥) ⊆ (𝑢𝑚 𝑥))
2725, 8, 26mp2an 679 . . . . . . . . . . . . 13 ((𝑢𝐴) ↑𝑚 𝑥) ⊆ (𝑢𝑚 𝑥)
28 ssralv 3923 . . . . . . . . . . . . 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 5078 . . . . . . . . . . . . . . . . 17 (𝑢𝐴) ∈ V
31 vex 3418 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
3230, 31elmap 8235 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ↔ 𝑦:𝑥⟶(𝑢𝐴))
33 fss 6357 . . . . . . . . . . . . . . . . 17 ((𝑦:𝑥⟶(𝑢𝐴) ∧ (𝑢𝐴) ⊆ 𝐴) → 𝑦:𝑥𝐴)
3411, 33mpan2 678 . . . . . . . . . . . . . . . 16 (𝑦:𝑥⟶(𝑢𝐴) → 𝑦:𝑥𝐴)
3532, 34sylbi 209 . . . . . . . . . . . . . . 15 (𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) → 𝑦:𝑥𝐴)
3635imim1i 63 . . . . . . . . . . . . . 14 ((𝑦:𝑥𝐴 ran 𝑦𝐴) → (𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) → ran 𝑦𝐴))
3736alimi 1774 . . . . . . . . . . . . 13 (∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴) → ∀𝑦(𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) → ran 𝑦𝐴))
38 df-ral 3093 . . . . . . . . . . . . 13 (∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦𝐴 ↔ ∀𝑦(𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) → ran 𝑦𝐴))
3937, 38sylibr 226 . . . . . . . . . . . 12 (∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴) → ∀𝑦 ∈ ((𝑢𝐴) ↑𝑚 𝑥) ran 𝑦𝐴)
40 elin 4057 . . . . . . . . . . . . . 14 ( ran 𝑦 ∈ (𝑢𝐴) ↔ ( ran 𝑦𝑢 ran 𝑦𝐴))
4140simplbi2 493 . . . . . . . . . . . . 13 ( ran 𝑦𝑢 → ( ran 𝑦𝐴 ran 𝑦 ∈ (𝑢𝐴)))
4241ral2imi 3106 . . . . . . . . . . . 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 610 . . . . . . . . . 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 1090 . . . . . . . . 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 1070 . . . . . . . . 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 1070 . . . . . . . . 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 288 . . . . . . . 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 3106 . . . . . . 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 610 . . . . 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 10012 . . . . 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 244 . . 3 (𝑢 ∈ Univ → ((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑢𝐴) ∈ Univ))
563, 55vtoclga 3493 . 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 198   ∧ wa 387   ∧ w3a 1068  ∀wal 1505   = wceq 1507   ∈ wcel 2050  ∀wral 3088  Vcvv 3415   ∩ cin 3828   ⊆ wss 3829  𝒫 cpw 4422  {cpr 4443  ∪ cuni 4712  Tr wtr 5030  ran crn 5408  ⟶wf 6184  (class class class)co 6976   ↑𝑚 cmap 8206  Univcgru 10010 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196  df-ov 6979  df-oprab 6980  df-mpo 6981  df-1st 7501  df-2nd 7502  df-map 8208  df-gru 10011 This theorem is referenced by:  wfgru  10036
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