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Theorem ingru 10225
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 4178 . . . . 5 (𝑢 = 𝑈 → (𝑢𝐴) = (𝑈𝐴))
21eleq1d 2894 . . . 4 (𝑢 = 𝑈 → ((𝑢𝐴) ∈ Univ ↔ (𝑈𝐴) ∈ Univ))
32imbi2d 342 . . 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 10202 . . . . . 6 (𝑢 ∈ Univ → (𝑢 ∈ Univ ↔ (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢))))
54ibi 268 . . . . 5 (𝑢 ∈ Univ → (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢)))
6 trin 5173 . . . . . . 7 ((Tr 𝑢 ∧ Tr 𝐴) → Tr (𝑢𝐴))
76ex 413 . . . . . 6 (Tr 𝑢 → (Tr 𝐴 → Tr (𝑢𝐴)))
8 inss1 4202 . . . . . . . 8 (𝑢𝐴) ⊆ 𝑢
9 ssralv 4030 . . . . . . . 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 4203 . . . . . . . 8 (𝑢𝐴) ⊆ 𝐴
12 ssralv 4030 . . . . . . . 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 4166 . . . . . . . . . . . . 13 (𝒫 𝑥 ∈ (𝑢𝐴) ↔ (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝐴))
1514simplbi2 501 . . . . . . . . . . . 12 (𝒫 𝑥𝑢 → (𝒫 𝑥𝐴 → 𝒫 𝑥 ∈ (𝑢𝐴)))
16 ssralv 4030 . . . . . . . . . . . . . 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 4030 . . . . . . . . . . . . . 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 4166 . . . . . . . . . . . . . . 15 ({𝑥, 𝑦} ∈ (𝑢𝐴) ↔ ({𝑥, 𝑦} ∈ 𝑢 ∧ {𝑥, 𝑦} ∈ 𝐴))
2120simplbi2 501 . . . . . . . . . . . . . 14 ({𝑥, 𝑦} ∈ 𝑢 → ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ∈ (𝑢𝐴)))
2221ral2imi 3153 . . . . . . . . . . . . 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 619 . . . . . . . . . . 11 ((𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢) → ((𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴))))
25 vex 3495 . . . . . . . . . . . . . 14 𝑢 ∈ V
26 mapss 8441 . . . . . . . . . . . . . 14 ((𝑢 ∈ V ∧ (𝑢𝐴) ⊆ 𝑢) → ((𝑢𝐴) ↑m 𝑥) ⊆ (𝑢m 𝑥))
2725, 8, 26mp2an 688 . . . . . . . . . . . . 13 ((𝑢𝐴) ↑m 𝑥) ⊆ (𝑢m 𝑥)
28 ssralv 4030 . . . . . . . . . . . . 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 5212 . . . . . . . . . . . . . . . . 17 (𝑢𝐴) ∈ V
31 vex 3495 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
3230, 31elmap 8424 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) ↔ 𝑦:𝑥⟶(𝑢𝐴))
33 fss 6520 . . . . . . . . . . . . . . . . 17 ((𝑦:𝑥⟶(𝑢𝐴) ∧ (𝑢𝐴) ⊆ 𝐴) → 𝑦:𝑥𝐴)
3411, 33mpan2 687 . . . . . . . . . . . . . . . 16 (𝑦:𝑥⟶(𝑢𝐴) → 𝑦:𝑥𝐴)
3532, 34sylbi 218 . . . . . . . . . . . . . . 15 (𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → 𝑦:𝑥𝐴)
3635imim1i 63 . . . . . . . . . . . . . 14 ((𝑦:𝑥𝐴 ran 𝑦𝐴) → (𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → ran 𝑦𝐴))
3736alimi 1803 . . . . . . . . . . . . 13 (∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴) → ∀𝑦(𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → ran 𝑦𝐴))
38 df-ral 3140 . . . . . . . . . . . . 13 (∀𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) ran 𝑦𝐴 ↔ ∀𝑦(𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → ran 𝑦𝐴))
3937, 38sylibr 235 . . . . . . . . . . . 12 (∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴) → ∀𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) ran 𝑦𝐴)
40 elin 4166 . . . . . . . . . . . . . 14 ( ran 𝑦 ∈ (𝑢𝐴) ↔ ( ran 𝑦𝑢 ran 𝑦𝐴))
4140simplbi2 501 . . . . . . . . . . . . 13 ( ran 𝑦𝑢 → ( ran 𝑦𝐴 ran 𝑦 ∈ (𝑢𝐴)))
4241ral2imi 3153 . . . . . . . . . . . 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 619 . . . . . . . . . 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 1102 . . . . . . . . 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 1081 . . . . . . . . 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 1081 . . . . . . . . 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 297 . . . . . . . 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 3153 . . . . . . 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 619 . . . . 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 10202 . . . . 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 253 . . 3 (𝑢 ∈ Univ → ((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑢𝐴) ∈ Univ))
563, 55vtoclga 3571 . 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 207  wa 396  w3a 1079  wal 1526   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cin 3932  wss 3933  𝒫 cpw 4535  {cpr 4559   cuni 4830  Tr wtr 5163  ran crn 5549  wf 6344  (class class class)co 7145  m cmap 8395  Univcgru 10200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-gru 10201
This theorem is referenced by:  wfgru  10226
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