Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brtrclfv2 Structured version   Visualization version   GIF version

Theorem brtrclfv2 42411
Description: Two ways to indicate two elements are related by the transitive closure of a relation. (Contributed by RP, 1-Jul-2020.)
Assertion
Ref Expression
brtrclfv2 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑌 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
Distinct variable groups:   𝑅,𝑓   𝑈,𝑓   𝑓,𝑉   𝑓,𝑊   𝑓,𝑋   𝑓,𝑌

Proof of Theorem brtrclfv2
Dummy variables 𝑔 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5148 . . . 4 (𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
21a1i 11 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}))
3 trclfv 14943 . . . . 5 (𝑅𝑊 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
43breqd 5158 . . . 4 (𝑅𝑊 → (𝑋(t+‘𝑅)𝑌𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌))
543ad2ant3 1136 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌))
6 elimasng 6084 . . . 4 ((𝑋𝑈𝑌𝑉) → (𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}))
763adant3 1133 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}))
82, 5, 73bitr4d 311 . 2 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋})))
9 intimasn 42341 . . . . 5 (𝑋𝑈 → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
1093ad2ant1 1134 . . . 4 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
11 simpl3 1194 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅𝑊)
12 snex 5430 . . . . . . . . . . . . . . 15 {𝑋} ∈ V
13 vex 3479 . . . . . . . . . . . . . . 15 𝑓 ∈ V
1412, 13xpex 7735 . . . . . . . . . . . . . 14 ({𝑋} × 𝑓) ∈ V
15 unexg 7731 . . . . . . . . . . . . . 14 ((𝑅𝑊 ∧ ({𝑋} × 𝑓) ∈ V) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
1611, 14, 15sylancl 587 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
17 trclfvlb 14951 . . . . . . . . . . . . . 14 ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
1817unssad 4186 . . . . . . . . . . . . 13 ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
1916, 18syl 17 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
20 trclfvcotrg 14959 . . . . . . . . . . . . 13 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))
2120a1i 11 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
22 simpl1 1192 . . . . . . . . . . . . . . . . 17 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋𝑈)
23 snidg 4661 . . . . . . . . . . . . . . . . 17 (𝑋𝑈𝑋 ∈ {𝑋})
2422, 23syl 17 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋 ∈ {𝑋})
25 inelcm 4463 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ {𝑋} ∧ 𝑋 ∈ {𝑋}) → ({𝑋} ∩ {𝑋}) ≠ ∅)
2624, 24, 25syl2anc 585 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} ∩ {𝑋}) ≠ ∅)
27 xpima2 6180 . . . . . . . . . . . . . . 15 (({𝑋} ∩ {𝑋}) ≠ ∅ → (({𝑋} × 𝑓) “ {𝑋}) = 𝑓)
2826, 27syl 17 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ {𝑋}) = 𝑓)
2916, 17syl 17 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
3029unssbd 4187 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
31 imass1 6097 . . . . . . . . . . . . . . 15 (({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) → (({𝑋} × 𝑓) “ {𝑋}) ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
3230, 31syl 17 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ {𝑋}) ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
3328, 32eqsstrrd 4020 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
34 imaundir 6147 . . . . . . . . . . . . . . 15 ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)))
35 simpr 486 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
36 imassrn 6068 . . . . . . . . . . . . . . . . . 18 (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ ran ({𝑋} × 𝑓)
37 rnxpss 6168 . . . . . . . . . . . . . . . . . 18 ran ({𝑋} × 𝑓) ⊆ 𝑓
3836, 37sstri 3990 . . . . . . . . . . . . . . . . 17 (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓
3938a1i 11 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
4035, 39unssd 4185 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓))) ⊆ 𝑓)
4134, 40eqsstrid 4029 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
42 trclimalb2 42410 . . . . . . . . . . . . . 14 (((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V ∧ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
4316, 41, 42syl2anc 585 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
4433, 43eqssd 3998 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
45 sbcan 3828 . . . . . . . . . . . . 13 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})) ↔ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋})))
46 sbcan 3828 . . . . . . . . . . . . . . 15 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟[(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟))
47 fvex 6901 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V
48 sbcssg 4522 . . . . . . . . . . . . . . . . . 18 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟))
4947, 48ax-mp 5 . . . . . . . . . . . . . . . . 17 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟)
50 csbconstg 3911 . . . . . . . . . . . . . . . . . . 19 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅 = 𝑅)
5147, 50ax-mp 5 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅 = 𝑅
5247csbvargi 4431 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))
5351, 52sseq12i 4011 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
5449, 53bitri 275 . . . . . . . . . . . . . . . 16 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
55 sbcssg 4522 . . . . . . . . . . . . . . . . . 18 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟))
5647, 55ax-mp 5 . . . . . . . . . . . . . . . . 17 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟)
57 csbcog 6293 . . . . . . . . . . . . . . . . . . . 20 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟))
5847, 57ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟)
5952, 52coeq12i 5861 . . . . . . . . . . . . . . . . . . 19 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6058, 59eqtri 2761 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6160, 52sseq12i 4011 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟 ↔ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6256, 61bitri 275 . . . . . . . . . . . . . . . 16 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟 ↔ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6354, 62anbi12i 628 . . . . . . . . . . . . . . 15 (([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟[(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟) ↔ (𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))))
6446, 63bitri 275 . . . . . . . . . . . . . 14 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ (𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))))
65 sbceq2g 4415 . . . . . . . . . . . . . . . 16 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋})))
6647, 65ax-mp 5 . . . . . . . . . . . . . . 15 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}))
67 csbima12 6075 . . . . . . . . . . . . . . . . 17 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋})
6852imaeq1i 6054 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋})
69 csbconstg 3911 . . . . . . . . . . . . . . . . . . 19 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋} = {𝑋})
7047, 69ax-mp 5 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋} = {𝑋}
7170imaeq2i 6055 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
7267, 68, 713eqtri 2765 . . . . . . . . . . . . . . . 16 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
7372eqeq2i 2746 . . . . . . . . . . . . . . 15 (𝑓 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
7466, 73bitri 275 . . . . . . . . . . . . . 14 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
7564, 74anbi12i 628 . . . . . . . . . . . . 13 (([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋})) ↔ ((𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ∧ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})))
7645, 75sylbbr 235 . . . . . . . . . . . 12 (((𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ∧ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})) → [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
7719, 21, 44, 76syl21anc 837 . . . . . . . . . . 11 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
7877spesbcd 3876 . . . . . . . . . 10 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
7978ex 414 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
80 eqeq1 2737 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑠 “ {𝑋})))
8180rexbidv 3179 . . . . . . . . . . 11 (𝑔 = 𝑓 → (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋})))
82 imaeq1 6052 . . . . . . . . . . . . 13 (𝑠 = 𝑟 → (𝑠 “ {𝑋}) = (𝑟 “ {𝑋}))
8382eqeq2d 2744 . . . . . . . . . . . 12 (𝑠 = 𝑟 → (𝑓 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑟 “ {𝑋})))
8483rexab2 3694 . . . . . . . . . . 11 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
8581, 84bitrdi 287 . . . . . . . . . 10 (𝑔 = 𝑓 → (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
8613, 85elab 3667 . . . . . . . . 9 (𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
8779, 86syl6ibr 252 . . . . . . . 8 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})}))
88 intss1 4966 . . . . . . . 8 (𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} → {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓)
8987, 88syl6 35 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
9089alrimiv 1931 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ∀𝑓((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
91 ssintab 4968 . . . . . 6 ( {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ↔ ∀𝑓((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
9290, 91sylibr 233 . . . . 5 ((𝑋𝑈𝑌𝑉𝑅𝑊) → {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
93 ssintab 4968 . . . . . . 7 ( {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ↔ ∀𝑔(∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔))
9482eqeq2d 2744 . . . . . . . . . 10 (𝑠 = 𝑟 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑔 = (𝑟 “ {𝑋})))
9594rexab2 3694 . . . . . . . . 9 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})))
96 simpr 486 . . . . . . . . . . 11 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 = (𝑟 “ {𝑋}))
97 imass1 6097 . . . . . . . . . . . . . . 15 (𝑅𝑟 → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
9897adantr 482 . . . . . . . . . . . . . 14 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
99 imass1 6097 . . . . . . . . . . . . . . 15 (𝑅𝑟 → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ (𝑟 “ {𝑋})))
100 imaco 6247 . . . . . . . . . . . . . . . 16 ((𝑟𝑟) “ {𝑋}) = (𝑟 “ (𝑟 “ {𝑋}))
101 imass1 6097 . . . . . . . . . . . . . . . 16 ((𝑟𝑟) ⊆ 𝑟 → ((𝑟𝑟) “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
102100, 101eqsstrrid 4030 . . . . . . . . . . . . . . 15 ((𝑟𝑟) ⊆ 𝑟 → (𝑟 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
10399, 102sylan9ss 3994 . . . . . . . . . . . . . 14 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
10498, 103jca 513 . . . . . . . . . . . . 13 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
105104adantr 482 . . . . . . . . . . . 12 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
106 vex 3479 . . . . . . . . . . . . . 14 𝑟 ∈ V
107106imaex 7902 . . . . . . . . . . . . 13 (𝑟 “ {𝑋}) ∈ V
108 imaundi 6146 . . . . . . . . . . . . . . . 16 (𝑅 “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ {𝑋}) ∪ (𝑅𝑓))
109108sseq1i 4009 . . . . . . . . . . . . . . 15 ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ∪ (𝑅𝑓)) ⊆ 𝑓)
110 unss 4183 . . . . . . . . . . . . . . 15 (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ∪ (𝑅𝑓)) ⊆ 𝑓)
111109, 110bitr4i 278 . . . . . . . . . . . . . 14 ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅𝑓) ⊆ 𝑓))
112 imaeq2 6053 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑟 “ {𝑋}) → (𝑅𝑓) = (𝑅 “ (𝑟 “ {𝑋})))
113 id 22 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑟 “ {𝑋}) → 𝑓 = (𝑟 “ {𝑋}))
114112, 113sseq12d 4014 . . . . . . . . . . . . . . 15 (𝑓 = (𝑟 “ {𝑋}) → ((𝑅𝑓) ⊆ 𝑓 ↔ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
115114cleq2lem 42292 . . . . . . . . . . . . . 14 (𝑓 = (𝑟 “ {𝑋}) → (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
116111, 115bitrid 283 . . . . . . . . . . . . 13 (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
117107, 116elab 3667 . . . . . . . . . . . 12 ((𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
118105, 117sylibr 233 . . . . . . . . . . 11 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → (𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
11996, 118eqeltrd 2834 . . . . . . . . . 10 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
120119exlimiv 1934 . . . . . . . . 9 (∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
12195, 120sylbi 216 . . . . . . . 8 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
122 intss1 4966 . . . . . . . 8 (𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} → {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
123121, 122syl 17 . . . . . . 7 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
12493, 123mpgbir 1802 . . . . . 6 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})}
125124a1i 11 . . . . 5 ((𝑋𝑈𝑌𝑉𝑅𝑊) → {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
12692, 125eqssd 3998 . . . 4 ((𝑋𝑈𝑌𝑉𝑅𝑊) → {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} = {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
12710, 126eqtrd 2773 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
128127eleq2d 2820 . 2 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ 𝑌 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
1298, 128bitrd 279 1 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑌 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wne 2941  wrex 3071  Vcvv 3475  [wsbc 3776  csb 3892  cun 3945  cin 3946  wss 3947  c0 4321  {csn 4627  cop 4633   cint 4949   class class class wbr 5147   × cxp 5673  ran crn 5676  cima 5678  ccom 5679  cfv 6540  t+ctcl 14928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-seq 13963  df-trcl 14930  df-relexp 14963
This theorem is referenced by:  dffrege76  42623
  Copyright terms: Public domain W3C validator