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

Theorem cnfinltrel 33578
Description: Less than for the Cantor normal form is a relation. (Contributed by ML, 24-Jun-2022.)
Hypotheses
Ref Expression
cnfin.1 𝐼 = {⟨∅, 1𝑜⟩}
cnfin.add + = (𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑛 ∈ (dom 𝑦 ∪ dom 𝑧) ↦ ((𝑦𝑛) +𝑜 (𝑧𝑛))))
cnfin.tr (𝜑 ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))
cnfin.ltadd (𝜓 ↔ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)))
cnfin.ltexp (𝜒 ↔ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩})))
cnfin.yrule 𝑌 = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝑐 ∨ (𝜑 ∨ (𝜓𝜒)))}
cnfin.lt < = ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)
cnfin.def 𝐶 = dom <
Assertion
Ref Expression
cnfinltrel Rel <
Distinct variable groups:   𝐼,𝑐   𝑎,𝑏,𝑐,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   𝜓(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   𝜒(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   𝐶(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   + (𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   < (𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)   𝐼(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏)   𝑌(𝑥,𝑦,𝑧,𝑛,𝑎,𝑏,𝑐)

Proof of Theorem cnfinltrel
Dummy variables 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 reluni 5380 . . 3 (Rel ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) ↔ ∀𝑘 ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)Rel 𝑘)
2 frfnom 7683 . . . . 5 (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) Fn ω
3 fvelrnb 6385 . . . . 5 ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) Fn ω → (𝑘 ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) ↔ ∃ ∈ ω ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = 𝑘))
42, 3ax-mp 5 . . . 4 (𝑘 ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) ↔ ∃ ∈ ω ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = 𝑘)
5 0ex 4924 . . . . . . . . 9 ∅ ∈ V
6 cnfin.1 . . . . . . . . . 10 𝐼 = {⟨∅, 1𝑜⟩}
7 snex 5036 . . . . . . . . . 10 {⟨∅, 1𝑜⟩} ∈ V
86, 7eqeltri 2846 . . . . . . . . 9 𝐼 ∈ V
95, 8relsnop 5367 . . . . . . . 8 Rel {⟨∅, 𝐼⟩}
10 fveq2 6332 . . . . . . . . . . 11 ( = ∅ → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘∅))
11 snex 5036 . . . . . . . . . . . 12 {⟨∅, 𝐼⟩} ∈ V
12 fr0g 7684 . . . . . . . . . . . 12 ({⟨∅, 𝐼⟩} ∈ V → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘∅) = {⟨∅, 𝐼⟩})
1311, 12ax-mp 5 . . . . . . . . . . 11 ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘∅) = {⟨∅, 𝐼⟩}
1410, 13syl6eq 2821 . . . . . . . . . 10 ( = ∅ → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = {⟨∅, 𝐼⟩})
1514releqd 5343 . . . . . . . . 9 ( = ∅ → (Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel {⟨∅, 𝐼⟩}))
165, 15sbcie 3622 . . . . . . . 8 ([∅ / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel {⟨∅, 𝐼⟩})
179, 16mpbir 221 . . . . . . 7 [∅ / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)
18 fvex 6342 . . . . . . . . . . . 12 ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ∈ V
19 cnfin.yrule . . . . . . . . . . . . . 14 𝑌 = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝑐 ∨ (𝜑 ∨ (𝜓𝜒)))}
2019relopabi 5384 . . . . . . . . . . . . 13 Rel 𝑌
2120sbcth 3602 . . . . . . . . . . . 12 (((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ∈ V → [((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐]Rel 𝑌)
2218, 21ax-mp 5 . . . . . . . . . . 11 [((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐]Rel 𝑌
23 sbcrel 5345 . . . . . . . . . . . 12 (((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ∈ V → ([((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐]Rel 𝑌 ↔ Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌))
2418, 23ax-mp 5 . . . . . . . . . . 11 ([((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐]Rel 𝑌 ↔ Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌)
2522, 24mpbi 220 . . . . . . . . . 10 Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌
26 frsuc 7685 . . . . . . . . . . . 12 ( ∈ ω → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ) = ((𝑐 ∈ V ↦ 𝑌)‘((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)))
27 cnfin.ltexp . . . . . . . . . . . . . . . . 17 (𝜒 ↔ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩})))
2827opabbii 4851 . . . . . . . . . . . . . . . 16 {⟨𝑥, 𝑦⟩ ∣ 𝜒} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))}
29 vex 3354 . . . . . . . . . . . . . . . . . . 19 𝑐 ∈ V
3029dmex 7246 . . . . . . . . . . . . . . . . . 18 dom 𝑐 ∈ V
3129rnex 7247 . . . . . . . . . . . . . . . . . 18 ran 𝑐 ∈ V
3230, 31ab2rexex 7306 . . . . . . . . . . . . . . . . 17 {𝑧 ∣ ∃𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩} ∈ V
33 df-opab 4847 . . . . . . . . . . . . . . . . . 18 {⟨𝑥, 𝑦⟩ ∣ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))} = {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩})))}
34 19.42vv 2035 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑎𝑏(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) ↔ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))))
35342exbii 1925 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑥𝑦𝑎𝑏(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) ↔ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))))
36 vex 3354 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑎 ∈ V
37 vex 3354 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑏 ∈ V
3836, 37opeldm 5466 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑎, 𝑏⟩ ∈ 𝑐𝑎 ∈ dom 𝑐)
3936, 37opelrn 5495 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑎, 𝑏⟩ ∈ 𝑐𝑏 ∈ ran 𝑐)
4038, 39jca 501 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨𝑎, 𝑏⟩ ∈ 𝑐 → (𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐))
4140ad2antrl 707 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → (𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐))
42 opeq12 4541 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}) → ⟨𝑥, 𝑦⟩ = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩)
4342eqeq2d 2781 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}) → (𝑧 = ⟨𝑥, 𝑦⟩ ↔ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
4443biimpac 464 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩})) → 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩)
4544adantrl 695 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩)
4641, 45jca 501 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → ((𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐) ∧ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
47462eximi 1911 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑎𝑏(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → ∃𝑎𝑏((𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐) ∧ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
4847exlimivv 2012 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑥𝑦𝑎𝑏(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → ∃𝑎𝑏((𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐) ∧ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
4935, 48sylbir 225 . . . . . . . . . . . . . . . . . . . 20 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → ∃𝑎𝑏((𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐) ∧ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
50 r2ex 3209 . . . . . . . . . . . . . . . . . . . 20 (∃𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩ ↔ ∃𝑎𝑏((𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐) ∧ 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩))
5149, 50sylibr 224 . . . . . . . . . . . . . . . . . . 19 (∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))) → ∃𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩)
5251ss2abi 3823 . . . . . . . . . . . . . . . . . 18 {𝑧 ∣ ∃𝑥𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩})))} ⊆ {𝑧 ∣ ∃𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩}
5333, 52eqsstri 3784 . . . . . . . . . . . . . . . . 17 {⟨𝑥, 𝑦⟩ ∣ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))} ⊆ {𝑧 ∣ ∃𝑎 ∈ dom 𝑐𝑏 ∈ ran 𝑐 𝑧 = ⟨{⟨𝑎, 1𝑜⟩}, {⟨𝑏, 1𝑜⟩}⟩}
5432, 53ssexi 4937 . . . . . . . . . . . . . . . 16 {⟨𝑥, 𝑦⟩ ∣ ∃𝑎𝑏(⟨𝑎, 𝑏⟩ ∈ 𝑐 ∧ (𝑥 = {⟨𝑎, 1𝑜⟩} ∧ 𝑦 = {⟨𝑏, 1𝑜⟩}))} ∈ V
5528, 54eqeltri 2846 . . . . . . . . . . . . . . 15 {⟨𝑥, 𝑦⟩ ∣ 𝜒} ∈ V
56 unopab 4862 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜒}) = {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)}
57 cnfin.ltadd . . . . . . . . . . . . . . . . . . . 20 (𝜓 ↔ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)))
5857opabbii 4851 . . . . . . . . . . . . . . . . . . 19 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏))}
5930, 31unex 7103 . . . . . . . . . . . . . . . . . . . 20 (dom 𝑐 ∪ ran 𝑐) ∈ V
60 nfcv 2913 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦ran 𝑐
61 nfcv 2913 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑦𝑥
62 cnfin.add . . . . . . . . . . . . . . . . . . . . . . . . . . 27 + = (𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑛 ∈ (dom 𝑦 ∪ dom 𝑧) ↦ ((𝑦𝑛) +𝑜 (𝑧𝑛))))
63 nfmpt21 6869 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦(𝑦 ∈ V, 𝑧 ∈ V ↦ (𝑛 ∈ (dom 𝑦 ∪ dom 𝑧) ↦ ((𝑦𝑛) +𝑜 (𝑧𝑛))))
6462, 63nfcxfr 2911 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑦 +
65 nfcv 2913 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑦𝑏
6661, 64, 65nfov 6821 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑦(𝑥 + 𝑏)
6766nfeq2 2929 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑦 𝑘 = (𝑥 + 𝑏)
6860, 67nfrex 3155 . . . . . . . . . . . . . . . . . . . . . . 23 𝑦𝑏 ∈ ran 𝑐 𝑘 = (𝑥 + 𝑏)
69 nfv 1995 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)
70 eqeq1 2775 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑦 → (𝑘 = (𝑥 + 𝑏) ↔ 𝑦 = (𝑥 + 𝑏)))
7170rexbidv 3200 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑦 → (∃𝑏 ∈ ran 𝑐 𝑘 = (𝑥 + 𝑏) ↔ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)))
7268, 69, 71cbvab 2895 . . . . . . . . . . . . . . . . . . . . . 22 {𝑘 ∣ ∃𝑏 ∈ ran 𝑐 𝑘 = (𝑥 + 𝑏)} = {𝑦 ∣ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)}
73 abrexexg 7287 . . . . . . . . . . . . . . . . . . . . . . 23 (ran 𝑐 ∈ V → {𝑘 ∣ ∃𝑏 ∈ ran 𝑐 𝑘 = (𝑥 + 𝑏)} ∈ V)
7431, 73ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 {𝑘 ∣ ∃𝑏 ∈ ran 𝑐 𝑘 = (𝑥 + 𝑏)} ∈ V
7572, 74eqeltrri 2847 . . . . . . . . . . . . . . . . . . . . 21 {𝑦 ∣ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)} ∈ V
7675a1i 11 . . . . . . . . . . . . . . . . . . . 20 (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) → {𝑦 ∣ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏)} ∈ V)
7759, 76opabex3 7293 . . . . . . . . . . . . . . . . . . 19 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (dom 𝑐 ∪ ran 𝑐) ∧ ∃𝑏 ∈ ran 𝑐 𝑦 = (𝑥 + 𝑏))} ∈ V
7858, 77eqeltri 2846 . . . . . . . . . . . . . . . . . 18 {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V
79 unexg 7106 . . . . . . . . . . . . . . . . . 18 (({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜒} ∈ V) → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜒}) ∈ V)
8078, 79mpan 670 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ 𝜒} ∈ V → ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ∪ {⟨𝑥, 𝑦⟩ ∣ 𝜒}) ∈ V)
8156, 80syl5eqelr 2855 . . . . . . . . . . . . . . . 16 ({⟨𝑥, 𝑦⟩ ∣ 𝜒} ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)} ∈ V)
82 unopab 4862 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))}
83 cnfin.tr . . . . . . . . . . . . . . . . . . . 20 (𝜑 ↔ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))
8483opabbii 4851 . . . . . . . . . . . . . . . . . . 19 {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)}
8530, 31xpex 7109 . . . . . . . . . . . . . . . . . . . 20 (dom 𝑐 × ran 𝑐) ∈ V
86 df-opab 4847 . . . . . . . . . . . . . . . . . . . . 21 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)} = {𝑙 ∣ ∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))}
87 eleq1 2838 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑙 = ⟨𝑥, 𝑦⟩ → (𝑙 ∈ (dom 𝑐 × ran 𝑐) ↔ ⟨𝑥, 𝑦⟩ ∈ (dom 𝑐 × ran 𝑐)))
8887biimprd 238 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ (dom 𝑐 × ran 𝑐) → 𝑙 ∈ (dom 𝑐 × ran 𝑐)))
89 vex 3354 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑥 ∈ V
90 vex 3354 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑧 ∈ V
9189, 90opeldm 5466 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑥, 𝑧⟩ ∈ 𝑐𝑥 ∈ dom 𝑐)
92 vex 3354 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑦 ∈ V
9390, 92opelrn 5495 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑧, 𝑦⟩ ∈ 𝑐𝑦 ∈ ran 𝑐)
94 opelxpi 5288 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ dom 𝑐𝑦 ∈ ran 𝑐) → ⟨𝑥, 𝑦⟩ ∈ (dom 𝑐 × ran 𝑐))
9591, 93, 94syl2an 583 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐) → ⟨𝑥, 𝑦⟩ ∈ (dom 𝑐 × ran 𝑐))
9695exlimiv 2010 . . . . . . . . . . . . . . . . . . . . . . . 24 (∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐) → ⟨𝑥, 𝑦⟩ ∈ (dom 𝑐 × ran 𝑐))
9788, 96impel 495 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑙 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)) → 𝑙 ∈ (dom 𝑐 × ran 𝑐))
9897exlimivv 2012 . . . . . . . . . . . . . . . . . . . . . 22 (∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)) → 𝑙 ∈ (dom 𝑐 × ran 𝑐))
9998abssi 3826 . . . . . . . . . . . . . . . . . . . . 21 {𝑙 ∣ ∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐))} ⊆ (dom 𝑐 × ran 𝑐)
10086, 99eqsstri 3784 . . . . . . . . . . . . . . . . . . . 20 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)} ⊆ (dom 𝑐 × ran 𝑐)
10185, 100ssexi 4937 . . . . . . . . . . . . . . . . . . 19 {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(⟨𝑥, 𝑧⟩ ∈ 𝑐 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑐)} ∈ V
10284, 101eqeltri 2846 . . . . . . . . . . . . . . . . . 18 {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V
103 unexg 7106 . . . . . . . . . . . . . . . . . 18 (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∈ V ∧ {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)} ∈ V) → ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)}) ∈ V)
104102, 103mpan 670 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)} ∈ V → ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)}) ∈ V)
10582, 104syl5eqelr 2855 . . . . . . . . . . . . . . . 16 ({⟨𝑥, 𝑦⟩ ∣ (𝜓𝜒)} ∈ V → {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))} ∈ V)
106 unopab 4862 . . . . . . . . . . . . . . . . . 18 ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))}) = {⟨𝑥, 𝑦⟩ ∣ (⟨𝑥, 𝑦⟩ ∈ 𝑐 ∨ (𝜑 ∨ (𝜓𝜒)))}
107106, 19eqtr4i 2796 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))}) = 𝑌
108 df-opab 4847 . . . . . . . . . . . . . . . . . . . 20 {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} = {𝑙 ∣ ∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑐)}
109 eleq1 2838 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑙 = ⟨𝑥, 𝑦⟩ → (𝑙𝑐 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑐))
110109biimprd 238 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑙 = ⟨𝑥, 𝑦⟩ → (⟨𝑥, 𝑦⟩ ∈ 𝑐𝑙𝑐))
111 id 22 . . . . . . . . . . . . . . . . . . . . . . 23 (⟨𝑥, 𝑦⟩ ∈ 𝑐 → ⟨𝑥, 𝑦⟩ ∈ 𝑐)
112110, 111impel 495 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑙 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑐) → 𝑙𝑐)
113112exlimivv 2012 . . . . . . . . . . . . . . . . . . . . 21 (∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑐) → 𝑙𝑐)
114113abssi 3826 . . . . . . . . . . . . . . . . . . . 20 {𝑙 ∣ ∃𝑥𝑦(𝑙 = ⟨𝑥, 𝑦⟩ ∧ ⟨𝑥, 𝑦⟩ ∈ 𝑐)} ⊆ 𝑐
115108, 114eqsstri 3784 . . . . . . . . . . . . . . . . . . 19 {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ⊆ 𝑐
11629, 115ssexi 4937 . . . . . . . . . . . . . . . . . 18 {⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∈ V
117 unexg 7106 . . . . . . . . . . . . . . . . . 18 (({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∈ V ∧ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))} ∈ V) → ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))}) ∈ V)
118116, 117mpan 670 . . . . . . . . . . . . . . . . 17 ({⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))} ∈ V → ({⟨𝑥, 𝑦⟩ ∣ ⟨𝑥, 𝑦⟩ ∈ 𝑐} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))}) ∈ V)
119107, 118syl5eqelr 2855 . . . . . . . . . . . . . . . 16 ({⟨𝑥, 𝑦⟩ ∣ (𝜑 ∨ (𝜓𝜒))} ∈ V → 𝑌 ∈ V)
12081, 105, 1193syl 18 . . . . . . . . . . . . . . 15 ({⟨𝑥, 𝑦⟩ ∣ 𝜒} ∈ V → 𝑌 ∈ V)
12155, 120ax-mp 5 . . . . . . . . . . . . . 14 𝑌 ∈ V
122121csbex 4927 . . . . . . . . . . . . 13 ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌 ∈ V
123 eqid 2771 . . . . . . . . . . . . . 14 (𝑐 ∈ V ↦ 𝑌) = (𝑐 ∈ V ↦ 𝑌)
124123fvmpts 6427 . . . . . . . . . . . . 13 ((((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ∈ V ∧ ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌 ∈ V) → ((𝑐 ∈ V ↦ 𝑌)‘((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)) = ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌)
12518, 122, 124mp2an 672 . . . . . . . . . . . 12 ((𝑐 ∈ V ↦ 𝑌)‘((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)) = ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌
12626, 125syl6eq 2821 . . . . . . . . . . 11 ( ∈ ω → ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ) = ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌)
127126releqd 5343 . . . . . . . . . 10 ( ∈ ω → (Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ) ↔ Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) / 𝑐𝑌))
12825, 127mpbiri 248 . . . . . . . . 9 ( ∈ ω → Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ))
129 vex 3354 . . . . . . . . . . . 12 ∈ V
130129sucex 7158 . . . . . . . . . . 11 suc ∈ V
131 sbcrel 5345 . . . . . . . . . . 11 (suc ∈ V → ([suc / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel suc / ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)))
132130, 131ax-mp 5 . . . . . . . . . 10 ([suc / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel suc / ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘))
133 csbfv 6374 . . . . . . . . . . 11 suc / ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc )
134133releqi 5342 . . . . . . . . . 10 (Rel suc / ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ))
135132, 134bitri 264 . . . . . . . . 9 ([suc / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘suc ))
136128, 135sylibr 224 . . . . . . . 8 ( ∈ ω → [suc / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘))
137136a1d 25 . . . . . . 7 ( ∈ ω → (Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) → [suc / ]Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘)))
13817, 137findes 7243 . . . . . 6 ( ∈ ω → Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘))
139 releq 5341 . . . . . 6 (((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = 𝑘 → (Rel ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) ↔ Rel 𝑘))
140138, 139syl5ibcom 235 . . . . 5 ( ∈ ω → (((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = 𝑘 → Rel 𝑘))
141140rexlimiv 3175 . . . 4 (∃ ∈ ω ((rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)‘) = 𝑘 → Rel 𝑘)
1424, 141sylbi 207 . . 3 (𝑘 ∈ ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω) → Rel 𝑘)
1431, 142mprgbir 3076 . 2 Rel ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)
144 cnfin.lt . . 3 < = ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω)
145144releqi 5342 . 2 (Rel < ↔ Rel ran (rec((𝑐 ∈ V ↦ 𝑌), {⟨∅, 𝐼⟩}) ↾ ω))
146143, 145mpbir 221 1 Rel <
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  wo 836   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wrex 3062  Vcvv 3351  [wsbc 3587  csb 3682  cun 3721  c0 4063  {csn 4316  cop 4322   cuni 4574  {copab 4846  cmpt 4863   × cxp 5247  dom cdm 5249  ran crn 5250  cres 5251  Rel wrel 5254  suc csuc 5868   Fn wfn 6026  cfv 6031  (class class class)co 6793  cmpt2 6795  ωcom 7212  reccrdg 7658  1𝑜c1o 7706   +𝑜 coa 7710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator