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Theorem brtrclfv2 38336
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 4686 . . . 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 13785 . . . . 5 (𝑅𝑊 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
43breqd 4696 . . . 4 (𝑅𝑊 → (𝑋(t+‘𝑅)𝑌𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌))
543ad2ant3 1104 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌))
6 elimasng 5526 . . . 4 ((𝑋𝑈𝑌𝑉) → (𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}))
763adant3 1101 . . 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 38266 . . . . 5 (𝑋𝑈 → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
1093ad2ant1 1102 . . . 4 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
11 simpl3 1086 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅𝑊)
12 snex 4938 . . . . . . . . . . . . . . 15 {𝑋} ∈ V
13 vex 3234 . . . . . . . . . . . . . . 15 𝑓 ∈ V
1412, 13xpex 7004 . . . . . . . . . . . . . 14 ({𝑋} × 𝑓) ∈ V
15 unexg 7001 . . . . . . . . . . . . . 14 ((𝑅𝑊 ∧ ({𝑋} × 𝑓) ∈ V) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
1611, 14, 15sylancl 695 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
17 trclfvlb 13793 . . . . . . . . . . . . . 14 ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
1817unssad 3823 . . . . . . . . . . . . 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 13801 . . . . . . . . . . . . 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 1084 . . . . . . . . . . . . . . . . 17 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋𝑈)
23 snidg 4239 . . . . . . . . . . . . . . . . 17 (𝑋𝑈𝑋 ∈ {𝑋})
2422, 23syl 17 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋 ∈ {𝑋})
25 inelcm 4065 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ {𝑋} ∧ 𝑋 ∈ {𝑋}) → ({𝑋} ∩ {𝑋}) ≠ ∅)
2624, 24, 25syl2anc 694 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} ∩ {𝑋}) ≠ ∅)
27 xpima2 5613 . . . . . . . . . . . . . . 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 3824 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
31 imass1 5535 . . . . . . . . . . . . . . 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 3673 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
34 imaundir 5581 . . . . . . . . . . . . . . 15 ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)))
35 simpr 476 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
36 imassrn 5512 . . . . . . . . . . . . . . . . . 18 (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ ran ({𝑋} × 𝑓)
37 rnxpss 5601 . . . . . . . . . . . . . . . . . 18 ran ({𝑋} × 𝑓) ⊆ 𝑓
3836, 37sstri 3645 . . . . . . . . . . . . . . . . 17 (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓
3938a1i 11 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
4035, 39unssd 3822 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓))) ⊆ 𝑓)
4134, 40syl5eqss 3682 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
42 trclimalb2 38335 . . . . . . . . . . . . . 14 (((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V ∧ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
4316, 41, 42syl2anc 694 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
4433, 43eqssd 3653 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
45 sbcan 3511 . . . . . . . . . . . . 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 3511 . . . . . . . . . . . . . . 15 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟[(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟))
47 fvex 6239 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V
48 sbcssg 4118 . . . . . . . . . . . . . . . . . 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 3579 . . . . . . . . . . . . . . . . . . 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 4036 . . . . . . . . . . . . . . . . . . 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 3664 . . . . . . . . . . . . . . . . 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 4118 . . . . . . . . . . . . . . . . . 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 38258 . . . . . . . . . . . . . . . . . . . 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 5318 . . . . . . . . . . . . . . . . . . 19 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6159, 60eqtri 2673 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6261, 53sseq12i 3664 . . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . 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 4023 . . . . . . . . . . . . . . . 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 5518 . . . . . . . . . . . . . . . . 17 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋})
6953imaeq1i 5498 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋})
70 csbconstg 3579 . . . . . . . . . . . . . . . . . . 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 5499 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
7368, 69, 723eqtri 2677 . . . . . . . . . . . . . . . 16 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
7473eqeq2i 2663 . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . 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 1365 . . . . . . . . . . 11 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
7978spesbcd 3555 . . . . . . . . . 10 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
8079ex 449 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
81 eqeq1 2655 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑠 “ {𝑋})))
8281rexbidv 3081 . . . . . . . . . . 11 (𝑔 = 𝑓 → (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋})))
83 imaeq1 5496 . . . . . . . . . . . . 13 (𝑠 = 𝑟 → (𝑠 “ {𝑋}) = (𝑟 “ {𝑋}))
8483eqeq2d 2661 . . . . . . . . . . . 12 (𝑠 = 𝑟 → (𝑓 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑟 “ {𝑋})))
8584rexab2 3406 . . . . . . . . . . 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 3382 . . . . . . . . 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 4524 . . . . . . . 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 1895 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ∀𝑓((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
92 ssintab 4526 . . . . . 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 4526 . . . . . . 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 2661 . . . . . . . . . 10 (𝑠 = 𝑟 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑔 = (𝑟 “ {𝑋})))
9695rexab2 3406 . . . . . . . . 9 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})))
97 simpr 476 . . . . . . . . . . 11 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 = (𝑟 “ {𝑋}))
98 imass1 5535 . . . . . . . . . . . . . . 15 (𝑅𝑟 → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
9998adantr 480 . . . . . . . . . . . . . 14 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
100 imass1 5535 . . . . . . . . . . . . . . 15 (𝑅𝑟 → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ (𝑟 “ {𝑋})))
101 imaco 5678 . . . . . . . . . . . . . . . 16 ((𝑟𝑟) “ {𝑋}) = (𝑟 “ (𝑟 “ {𝑋}))
102 imass1 5535 . . . . . . . . . . . . . . . 16 ((𝑟𝑟) ⊆ 𝑟 → ((𝑟𝑟) “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
103101, 102syl5eqssr 3683 . . . . . . . . . . . . . . 15 ((𝑟𝑟) ⊆ 𝑟 → (𝑟 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
104100, 103sylan9ss 3649 . . . . . . . . . . . . . 14 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
10599, 104jca 553 . . . . . . . . . . . . 13 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
106105adantr 480 . . . . . . . . . . . 12 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
107 vex 3234 . . . . . . . . . . . . . 14 𝑟 ∈ V
108 imaexg 7145 . . . . . . . . . . . . . 14 (𝑟 ∈ V → (𝑟 “ {𝑋}) ∈ V)
109107, 108ax-mp 5 . . . . . . . . . . . . 13 (𝑟 “ {𝑋}) ∈ V
110 imaundi 5580 . . . . . . . . . . . . . . . 16 (𝑅 “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ {𝑋}) ∪ (𝑅𝑓))
111110sseq1i 3662 . . . . . . . . . . . . . . 15 ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ∪ (𝑅𝑓)) ⊆ 𝑓)
112 unss 3820 . . . . . . . . . . . . . . 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 5497 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑟 “ {𝑋}) → (𝑅𝑓) = (𝑅 “ (𝑟 “ {𝑋})))
115 id 22 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑟 “ {𝑋}) → 𝑓 = (𝑟 “ {𝑋}))
116114, 115sseq12d 3667 . . . . . . . . . . . . . . 15 (𝑓 = (𝑟 “ {𝑋}) → ((𝑅𝑓) ⊆ 𝑓 ↔ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
117116cleq2lem 38231 . . . . . . . . . . . . . 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 3382 . . . . . . . . . . . 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 2730 . . . . . . . . . 10 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
122121exlimiv 1898 . . . . . . . . 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 4524 . . . . . . . 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 1766 . . . . . 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 3653 . . . 4 ((𝑋𝑈𝑌𝑉𝑅𝑊) → {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} = {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
12910, 128eqtrd 2685 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
130129eleq2d 2716 . 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 383  w3a 1054  wal 1521   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  wrex 2942  Vcvv 3231  [wsbc 3468  csb 3566  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  cop 4216   cint 4507   class class class wbr 4685   × cxp 5141  ran crn 5144  cima 5146  ccom 5147  cfv 5926  t+ctcl 13770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-seq 12842  df-trcl 13772  df-relexp 13805
This theorem is referenced by:  dffrege76  38550
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