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Theorem brtrclfv2 37527
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 4619 . . . 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 13682 . . . . 5 (𝑅𝑊 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
43breqd 4629 . . . 4 (𝑅𝑊 → (𝑋(t+‘𝑅)𝑌𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌))
543ad2ant3 1082 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌))
6 elimasng 5455 . . . 4 ((𝑋𝑈𝑌𝑉) → (𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}))
763adant3 1079 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}))
82, 5, 73bitr4d 300 . 2 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋})))
9 intimasn 37457 . . . . 5 (𝑋𝑈 → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
1093ad2ant1 1080 . . . 4 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
11 simpl3 1064 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅𝑊)
12 snex 4874 . . . . . . . . . . . . . . 15 {𝑋} ∈ V
13 vex 3192 . . . . . . . . . . . . . . 15 𝑓 ∈ V
1412, 13xpex 6922 . . . . . . . . . . . . . 14 ({𝑋} × 𝑓) ∈ V
15 unexg 6919 . . . . . . . . . . . . . 14 ((𝑅𝑊 ∧ ({𝑋} × 𝑓) ∈ V) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
1611, 14, 15sylancl 693 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
17 trclfvlb 13690 . . . . . . . . . . . . . 14 ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
1817unssad 3773 . . . . . . . . . . . . 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 13698 . . . . . . . . . . . . 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 1062 . . . . . . . . . . . . . . . . 17 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋𝑈)
23 snidg 4182 . . . . . . . . . . . . . . . . 17 (𝑋𝑈𝑋 ∈ {𝑋})
2422, 23syl 17 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋 ∈ {𝑋})
25 inelcm 4009 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ {𝑋} ∧ 𝑋 ∈ {𝑋}) → ({𝑋} ∩ {𝑋}) ≠ ∅)
2624, 24, 25syl2anc 692 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} ∩ {𝑋}) ≠ ∅)
27 xpima2 5542 . . . . . . . . . . . . . . 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 3774 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
31 imass1 5464 . . . . . . . . . . . . . . 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, 32eqsstr3d 3624 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
34 imaundir 5510 . . . . . . . . . . . . . . 15 ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)))
35 simpr 477 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
36 imassrn 5441 . . . . . . . . . . . . . . . . . 18 (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ ran ({𝑋} × 𝑓)
37 rnxpss 5530 . . . . . . . . . . . . . . . . . 18 ran ({𝑋} × 𝑓) ⊆ 𝑓
3836, 37sstri 3596 . . . . . . . . . . . . . . . . 17 (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓
3938a1i 11 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
4035, 39unssd 3772 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓))) ⊆ 𝑓)
4134, 40syl5eqss 3633 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
42 trclimalb2 37526 . . . . . . . . . . . . . 14 (((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V ∧ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
4316, 41, 42syl2anc 692 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
4433, 43eqssd 3604 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
45 sbcan 3464 . . . . . . . . . . . . 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 3464 . . . . . . . . . . . . . . 15 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟[(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟))
47 fvex 6163 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V
48 sbcssg 4062 . . . . . . . . . . . . . . . . . 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 3531 . . . . . . . . . . . . . . . . . . 19 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅 = 𝑅)
5147, 50ax-mp 5 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅 = 𝑅
52 csbvarg 3980 . . . . . . . . . . . . . . . . . . 19 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
5347, 52ax-mp 5 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))
5451, 53sseq12i 3615 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
5549, 54bitri 264 . . . . . . . . . . . . . . . 16 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
56 sbcssg 4062 . . . . . . . . . . . . . . . . . 18 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟))
5747, 56ax-mp 5 . . . . . . . . . . . . . . . . 17 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟)
58 csbcog 37449 . . . . . . . . . . . . . . . . . . . 20 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟))
5947, 58ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟)
6053, 53coeq12i 5250 . . . . . . . . . . . . . . . . . . 19 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6159, 60eqtri 2643 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6261, 53sseq12i 3615 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟 ↔ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6357, 62bitri 264 . . . . . . . . . . . . . . . 16 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟 ↔ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6455, 63anbi12i 732 . . . . . . . . . . . . . . 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)))))
6546, 64bitri 264 . . . . . . . . . . . . . 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)))))
66 sbceq2g 3967 . . . . . . . . . . . . . . . 16 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋})))
6747, 66ax-mp 5 . . . . . . . . . . . . . . 15 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}))
68 csbima12 5447 . . . . . . . . . . . . . . . . 17 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋})
6953imaeq1i 5427 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋})
70 csbconstg 3531 . . . . . . . . . . . . . . . . . . 19 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋} = {𝑋})
7147, 70ax-mp 5 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋} = {𝑋}
7271imaeq2i 5428 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
7368, 69, 723eqtri 2647 . . . . . . . . . . . . . . . 16 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
7473eqeq2i 2633 . . . . . . . . . . . . . . 15 (𝑓 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
7567, 74bitri 264 . . . . . . . . . . . . . 14 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
7665, 75anbi12i 732 . . . . . . . . . . . . 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})))
7745, 76sylbbr 226 . . . . . . . . . . . 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})))
7819, 21, 44, 77syl21anc 1322 . . . . . . . . . . 11 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
7978spesbcd 3507 . . . . . . . . . 10 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
8079ex 450 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
81 eqeq1 2625 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑠 “ {𝑋})))
8281rexbidv 3046 . . . . . . . . . . 11 (𝑔 = 𝑓 → (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋})))
83 imaeq1 5425 . . . . . . . . . . . . 13 (𝑠 = 𝑟 → (𝑠 “ {𝑋}) = (𝑟 “ {𝑋}))
8483eqeq2d 2631 . . . . . . . . . . . 12 (𝑠 = 𝑟 → (𝑓 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑟 “ {𝑋})))
8584rexab2 3359 . . . . . . . . . . 11 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
8682, 85syl6bb 276 . . . . . . . . . 10 (𝑔 = 𝑓 → (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
8713, 86elab 3337 . . . . . . . . 9 (𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
8880, 87syl6ibr 242 . . . . . . . 8 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})}))
89 intss1 4462 . . . . . . . 8 (𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} → {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓)
9088, 89syl6 35 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
9190alrimiv 1852 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ∀𝑓((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
92 ssintab 4464 . . . . . 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))
9391, 92sylibr 224 . . . . 5 ((𝑋𝑈𝑌𝑉𝑅𝑊) → {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
94 ssintab 4464 . . . . . . 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))
9583eqeq2d 2631 . . . . . . . . . 10 (𝑠 = 𝑟 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑔 = (𝑟 “ {𝑋})))
9695rexab2 3359 . . . . . . . . 9 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})))
97 simpr 477 . . . . . . . . . . 11 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 = (𝑟 “ {𝑋}))
98 imass1 5464 . . . . . . . . . . . . . . 15 (𝑅𝑟 → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
9998adantr 481 . . . . . . . . . . . . . 14 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
100 imass1 5464 . . . . . . . . . . . . . . 15 (𝑅𝑟 → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ (𝑟 “ {𝑋})))
101 imaco 5604 . . . . . . . . . . . . . . . 16 ((𝑟𝑟) “ {𝑋}) = (𝑟 “ (𝑟 “ {𝑋}))
102 imass1 5464 . . . . . . . . . . . . . . . 16 ((𝑟𝑟) ⊆ 𝑟 → ((𝑟𝑟) “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
103101, 102syl5eqssr 3634 . . . . . . . . . . . . . . 15 ((𝑟𝑟) ⊆ 𝑟 → (𝑟 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
104100, 103sylan9ss 3600 . . . . . . . . . . . . . 14 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
10599, 104jca 554 . . . . . . . . . . . . 13 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
106105adantr 481 . . . . . . . . . . . 12 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
107 vex 3192 . . . . . . . . . . . . . 14 𝑟 ∈ V
108 imaexg 7057 . . . . . . . . . . . . . 14 (𝑟 ∈ V → (𝑟 “ {𝑋}) ∈ V)
109107, 108ax-mp 5 . . . . . . . . . . . . 13 (𝑟 “ {𝑋}) ∈ V
110 imaundi 5509 . . . . . . . . . . . . . . . 16 (𝑅 “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ {𝑋}) ∪ (𝑅𝑓))
111110sseq1i 3613 . . . . . . . . . . . . . . 15 ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ∪ (𝑅𝑓)) ⊆ 𝑓)
112 unss 3770 . . . . . . . . . . . . . . 15 (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ∪ (𝑅𝑓)) ⊆ 𝑓)
113111, 112bitr4i 267 . . . . . . . . . . . . . 14 ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅𝑓) ⊆ 𝑓))
114 imaeq2 5426 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑟 “ {𝑋}) → (𝑅𝑓) = (𝑅 “ (𝑟 “ {𝑋})))
115 id 22 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑟 “ {𝑋}) → 𝑓 = (𝑟 “ {𝑋}))
116114, 115sseq12d 3618 . . . . . . . . . . . . . . 15 (𝑓 = (𝑟 “ {𝑋}) → ((𝑅𝑓) ⊆ 𝑓 ↔ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
117116cleq2lem 37422 . . . . . . . . . . . . . 14 (𝑓 = (𝑟 “ {𝑋}) → (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
118113, 117syl5bb 272 . . . . . . . . . . . . 13 (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
119109, 118elab 3337 . . . . . . . . . . . 12 ((𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
120106, 119sylibr 224 . . . . . . . . . . 11 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → (𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
12197, 120eqeltrd 2698 . . . . . . . . . 10 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
122121exlimiv 1855 . . . . . . . . 9 (∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
12396, 122sylbi 207 . . . . . . . 8 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
124 intss1 4462 . . . . . . . 8 (𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} → {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
125123, 124syl 17 . . . . . . 7 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
12694, 125mpgbir 1723 . . . . . 6 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})}
127126a1i 11 . . . . 5 ((𝑋𝑈𝑌𝑉𝑅𝑊) → {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
12893, 127eqssd 3604 . . . 4 ((𝑋𝑈𝑌𝑉𝑅𝑊) → {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} = {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
12910, 128eqtrd 2655 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
130129eleq2d 2684 . 2 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ 𝑌 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
1318, 130bitrd 268 1 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑌 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wrex 2908  Vcvv 3189  [wsbc 3421  csb 3518  cun 3557  cin 3558  wss 3559  c0 3896  {csn 4153  cop 4159   cint 4445   class class class wbr 4618   × cxp 5077  ran crn 5080  cima 5082  ccom 5083  cfv 5852  t+ctcl 13665
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-n0 11244  df-z 11329  df-uz 11639  df-seq 12749  df-trcl 13667  df-relexp 13702
This theorem is referenced by:  dffrege76  37742
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