Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ingru Structured version   Visualization version   GIF version

Theorem ingru 10235
 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 4166 . . . . 5 (𝑢 = 𝑈 → (𝑢𝐴) = (𝑈𝐴))
21eleq1d 2900 . . . 4 (𝑢 = 𝑈 → ((𝑢𝐴) ∈ Univ ↔ (𝑈𝐴) ∈ Univ))
32imbi2d 344 . . 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 10212 . . . . . 6 (𝑢 ∈ Univ → (𝑢 ∈ Univ ↔ (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢))))
54ibi 270 . . . . 5 (𝑢 ∈ Univ → (Tr 𝑢 ∧ ∀𝑥𝑢 (𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ∧ ∀𝑦 ∈ (𝑢m 𝑥) ran 𝑦𝑢)))
6 trin 5168 . . . . . . 7 ((Tr 𝑢 ∧ Tr 𝐴) → Tr (𝑢𝐴))
76ex 416 . . . . . 6 (Tr 𝑢 → (Tr 𝐴 → Tr (𝑢𝐴)))
8 inss1 4190 . . . . . . . 8 (𝑢𝐴) ⊆ 𝑢
9 ssralv 4019 . . . . . . . 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 4191 . . . . . . . 8 (𝑢𝐴) ⊆ 𝐴
12 ssralv 4019 . . . . . . . 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 3935 . . . . . . . . . . . . 13 (𝒫 𝑥 ∈ (𝑢𝐴) ↔ (𝒫 𝑥𝑢 ∧ 𝒫 𝑥𝐴))
1514simplbi2 504 . . . . . . . . . . . 12 (𝒫 𝑥𝑢 → (𝒫 𝑥𝐴 → 𝒫 𝑥 ∈ (𝑢𝐴)))
16 ssralv 4019 . . . . . . . . . . . . . 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 4019 . . . . . . . . . . . . . 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 3935 . . . . . . . . . . . . . . 15 ({𝑥, 𝑦} ∈ (𝑢𝐴) ↔ ({𝑥, 𝑦} ∈ 𝑢 ∧ {𝑥, 𝑦} ∈ 𝐴))
2120simplbi2 504 . . . . . . . . . . . . . 14 ({𝑥, 𝑦} ∈ 𝑢 → ({𝑥, 𝑦} ∈ 𝐴 → {𝑥, 𝑦} ∈ (𝑢𝐴)))
2221ral2imi 3151 . . . . . . . . . . . . 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 622 . . . . . . . . . . 11 ((𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢) → ((𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑢𝐴) ∧ ∀𝑦 ∈ (𝑢𝐴){𝑥, 𝑦} ∈ (𝑢𝐴))))
25 vex 3483 . . . . . . . . . . . . . 14 𝑢 ∈ V
26 mapss 8449 . . . . . . . . . . . . . 14 ((𝑢 ∈ V ∧ (𝑢𝐴) ⊆ 𝑢) → ((𝑢𝐴) ↑m 𝑥) ⊆ (𝑢m 𝑥))
2725, 8, 26mp2an 691 . . . . . . . . . . . . 13 ((𝑢𝐴) ↑m 𝑥) ⊆ (𝑢m 𝑥)
28 ssralv 4019 . . . . . . . . . . . . 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 5207 . . . . . . . . . . . . . . . . 17 (𝑢𝐴) ∈ V
31 vex 3483 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
3230, 31elmap 8431 . . . . . . . . . . . . . . . 16 (𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) ↔ 𝑦:𝑥⟶(𝑢𝐴))
33 fss 6517 . . . . . . . . . . . . . . . . 17 ((𝑦:𝑥⟶(𝑢𝐴) ∧ (𝑢𝐴) ⊆ 𝐴) → 𝑦:𝑥𝐴)
3411, 33mpan2 690 . . . . . . . . . . . . . . . 16 (𝑦:𝑥⟶(𝑢𝐴) → 𝑦:𝑥𝐴)
3532, 34sylbi 220 . . . . . . . . . . . . . . 15 (𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → 𝑦:𝑥𝐴)
3635imim1i 63 . . . . . . . . . . . . . 14 ((𝑦:𝑥𝐴 ran 𝑦𝐴) → (𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → ran 𝑦𝐴))
3736alimi 1813 . . . . . . . . . . . . 13 (∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴) → ∀𝑦(𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → ran 𝑦𝐴))
38 df-ral 3138 . . . . . . . . . . . . 13 (∀𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) ran 𝑦𝐴 ↔ ∀𝑦(𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) → ran 𝑦𝐴))
3937, 38sylibr 237 . . . . . . . . . . . 12 (∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴) → ∀𝑦 ∈ ((𝑢𝐴) ↑m 𝑥) ran 𝑦𝐴)
40 elin 3935 . . . . . . . . . . . . . 14 ( ran 𝑦 ∈ (𝑢𝐴) ↔ ( ran 𝑦𝑢 ran 𝑦𝐴))
4140simplbi2 504 . . . . . . . . . . . . 13 ( ran 𝑦𝑢 → ( ran 𝑦𝐴 ran 𝑦 ∈ (𝑢𝐴)))
4241ral2imi 3151 . . . . . . . . . . . 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 622 . . . . . . . . . 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 1107 . . . . . . . . 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 1086 . . . . . . . . 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 1086 . . . . . . . . 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 299 . . . . . . . 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 3151 . . . . . . 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 622 . . . . 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 10212 . . . . 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 255 . . 3 (𝑢 ∈ Univ → ((Tr 𝐴 ∧ ∀𝑥𝐴 (𝒫 𝑥𝐴 ∧ ∀𝑦𝐴 {𝑥, 𝑦} ∈ 𝐴 ∧ ∀𝑦(𝑦:𝑥𝐴 ran 𝑦𝐴))) → (𝑢𝐴) ∈ Univ))
563, 55vtoclga 3560 . 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 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2115  ∀wral 3133  Vcvv 3480   ∩ cin 3918   ⊆ wss 3919  𝒫 cpw 4522  {cpr 4552  ∪ cuni 4824  Tr wtr 5158  ran crn 5543  ⟶wf 6339  (class class class)co 7149   ↑m cmap 8402  Univcgru 10210 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-map 8404  df-gru 10211 This theorem is referenced by:  wfgru  10236
 Copyright terms: Public domain W3C validator