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Theorem ingru 10572
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 4145 . . . . 5 (𝑢 = 𝑈 → (𝑢𝐴) = (𝑈𝐴))
21eleq1d 2825 . . . 4 (𝑢 = 𝑈 → ((𝑢𝐴) ∈ Univ ↔ (𝑈𝐴) ∈ Univ))
32imbi2d 341 . . 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 10549 . . . . . 6 (𝑢 ∈ Univ → (𝑢 ∈ Univ ↔ (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢))))
54ibi 266 . . . . 5 (𝑢 ∈ Univ → (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢)))
6 trin 5206 . . . . . . 7 ((Tr 𝑢 ∧ Tr 𝐴) → Tr (𝑢𝐴))
76ex 413 . . . . . 6 (Tr 𝑢 → (Tr 𝐴 → Tr (𝑢𝐴)))
8 inss1 4168 . . . . . . . 8 (𝑢𝐴) ⊆ 𝑢
9 ssralv 3992 . . . . . . . 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 4169 . . . . . . . 8 (𝑢𝐴) ⊆ 𝐴
12 ssralv 3992 . . . . . . . 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 3908 . . . . . . . . . . . . 13 (𝒫 𝑥 ∈ (𝑢𝐴) ↔ (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝐴))
1514simplbi2 501 . . . . . . . . . . . 12 (𝒫 𝑥𝑢 → (𝒫 𝑥𝐴 → 𝒫 𝑥 ∈ (𝑢𝐴)))
16 ssralv 3992 . . . . . . . . . . . . . 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 3992 . . . . . . . . . . . . . 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 3908 . . . . . . . . . . . . . . 15 ({𝑥, 𝑦} ∈ (𝑢𝐴) ↔ ({𝑥, 𝑦} ∈ 𝑢 ∧ {𝑥, 𝑦} ∈ 𝐴))
2120simplbi2 501 . . . . . . . . . . . . . 14 ({𝑥, 𝑦} ∈ 𝑢 → ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ∈ (𝑢𝐴)))
2221ral2imi 3084 . . . . . . . . . . . . 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 620 . . . . . . . . . . 11 ((𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢) → ((𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴))))
25 vex 3435 . . . . . . . . . . . . . 14 𝑢 ∈ V
26 mapss 8660 . . . . . . . . . . . . . 14 ((𝑢 ∈ V ∧ (𝑢𝐴) ⊆ 𝑢) → ((𝑢𝐴) ↑m 𝑥) ⊆ (𝑢m 𝑥))
2725, 8, 26mp2an 689 . . . . . . . . . . . . 13 ((𝑢𝐴) ↑m 𝑥) ⊆ (𝑢m 𝑥)
28 ssralv 3992 . . . . . . . . . . . . 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 5245 . . . . . . . . . . . . . . . . 17 (𝑢𝐴) ∈ V
31 vex 3435 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
3230, 31elmap 8642 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) ↔ 𝑦:𝑥⟶(𝑢𝐴))
33 fss 6615 . . . . . . . . . . . . . . . . 17 ((𝑦:𝑥⟶(𝑢𝐴) ∧ (𝑢𝐴) ⊆ 𝐴) → 𝑦:𝑥𝐴)
3411, 33mpan2 688 . . . . . . . . . . . . . . . 16 (𝑦:𝑥⟶(𝑢𝐴) → 𝑦:𝑥𝐴)
3532, 34sylbi 216 . . . . . . . . . . . . . . 15 (𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → 𝑦:𝑥𝐴)
3635imim1i 63 . . . . . . . . . . . . . 14 ((𝑦:𝑥𝐴 ran 𝑦𝐴) → (𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → ran 𝑦𝐴))
3736alimi 1818 . . . . . . . . . . . . 13 (∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴) → ∀𝑦(𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → ran 𝑦𝐴))
38 df-ral 3071 . . . . . . . . . . . . 13 (∀𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) ran 𝑦𝐴 ↔ ∀𝑦(𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → ran 𝑦𝐴))
3937, 38sylibr 233 . . . . . . . . . . . 12 (∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴) → ∀𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) ran 𝑦𝐴)
40 elin 3908 . . . . . . . . . . . . . 14 ( ran 𝑦 ∈ (𝑢𝐴) ↔ ( ran 𝑦𝑢 ran 𝑦𝐴))
4140simplbi2 501 . . . . . . . . . . . . 13 ( ran 𝑦𝑢 → ( ran 𝑦𝐴 ran 𝑦 ∈ (𝑢𝐴)))
4241ral2imi 3084 . . . . . . . . . . . 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 620 . . . . . . . . . 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 1109 . . . . . . . . 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 1088 . . . . . . . . 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 1088 . . . . . . . . 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 296 . . . . . . . 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 3084 . . . . . . 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 620 . . . . 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 10549 . . . . 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 251 . . 3 (𝑢 ∈ Univ → ((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑢𝐴) ∈ Univ))
563, 55vtoclga 3512 . 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 205  wa 396  w3a 1086  wal 1540   = wceq 1542  wcel 2110  wral 3066  Vcvv 3431  cin 3891  wss 3892  𝒫 cpw 4539  {cpr 4569   cuni 4845  Tr wtr 5196  ran crn 5591  wf 6428  (class class class)co 7271  m cmap 8598  Univcgru 10547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-fv 6440  df-ov 7274  df-oprab 7275  df-mpo 7276  df-1st 7824  df-2nd 7825  df-map 8600  df-gru 10548
This theorem is referenced by:  wfgru  10573
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