MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ackbij2 Structured version   Visualization version   GIF version

Theorem ackbij2 9657
Description: The Ackermann bijection, part 2: hereditarily finite sets can be represented by recursive binary notation. (Contributed by Stefan O'Rear, 18-Nov-2014.)
Hypotheses
Ref Expression
ackbij.f 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
ackbij.g 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))
ackbij.h 𝐻 = (rec(𝐺, ∅) “ ω)
Assertion
Ref Expression
ackbij2 𝐻: (𝑅1 “ ω)–1-1-onto→ω
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦

Proof of Theorem ackbij2
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6666 . . . . . 6 (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
2 fvex 6679 . . . . . 6 (rec(𝐺, ∅)‘𝑎) ∈ V
31, 2f1iunw 7636 . . . . 5 (∀𝑎 ∈ ω ((rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1→ω ∧ ∀𝑏 ∈ ω ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))) → 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎): 𝑎 ∈ ω (𝑅1𝑎)–1-1→ω)
4 ackbij.f . . . . . . . . 9 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
5 ackbij.g . . . . . . . . 9 𝐺 = (𝑥 ∈ V ↦ (𝑦 ∈ 𝒫 dom 𝑥 ↦ (𝐹‘(𝑥𝑦))))
64, 5ackbij2lem2 9654 . . . . . . . 8 (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1-onto→(card‘(𝑅1𝑎)))
7 f1of1 6610 . . . . . . . 8 ((rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1-onto→(card‘(𝑅1𝑎)) → (rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1→(card‘(𝑅1𝑎)))
86, 7syl 17 . . . . . . 7 (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1→(card‘(𝑅1𝑎)))
9 ordom 7580 . . . . . . . 8 Ord ω
10 r1fin 9194 . . . . . . . . 9 (𝑎 ∈ ω → (𝑅1𝑎) ∈ Fin)
11 ficardom 9382 . . . . . . . . 9 ((𝑅1𝑎) ∈ Fin → (card‘(𝑅1𝑎)) ∈ ω)
1210, 11syl 17 . . . . . . . 8 (𝑎 ∈ ω → (card‘(𝑅1𝑎)) ∈ ω)
13 ordelss 6204 . . . . . . . 8 ((Ord ω ∧ (card‘(𝑅1𝑎)) ∈ ω) → (card‘(𝑅1𝑎)) ⊆ ω)
149, 12, 13sylancr 587 . . . . . . 7 (𝑎 ∈ ω → (card‘(𝑅1𝑎)) ⊆ ω)
15 f1ss 6576 . . . . . . 7 (((rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1→(card‘(𝑅1𝑎)) ∧ (card‘(𝑅1𝑎)) ⊆ ω) → (rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1→ω)
168, 14, 15syl2anc 584 . . . . . 6 (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1→ω)
17 nnord 7579 . . . . . . . . 9 (𝑎 ∈ ω → Ord 𝑎)
18 nnord 7579 . . . . . . . . 9 (𝑏 ∈ ω → Ord 𝑏)
19 ordtri2or2 6284 . . . . . . . . 9 ((Ord 𝑎 ∧ Ord 𝑏) → (𝑎𝑏𝑏𝑎))
2017, 18, 19syl2an 595 . . . . . . . 8 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎𝑏𝑏𝑎))
214, 5ackbij2lem4 9656 . . . . . . . . . . 11 (((𝑏 ∈ ω ∧ 𝑎 ∈ ω) ∧ 𝑎𝑏) → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏))
2221ex 413 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ 𝑎 ∈ ω) → (𝑎𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
2322ancoms 459 . . . . . . . . 9 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
244, 5ackbij2lem4 9656 . . . . . . . . . 10 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏𝑎) → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))
2524ex 413 . . . . . . . . 9 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑏𝑎 → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
2623, 25orim12d 960 . . . . . . . 8 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((𝑎𝑏𝑏𝑎) → ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))))
2720, 26mpd 15 . . . . . . 7 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
2827ralrimiva 3186 . . . . . 6 (𝑎 ∈ ω → ∀𝑏 ∈ ω ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
2916, 28jca 512 . . . . 5 (𝑎 ∈ ω → ((rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1→ω ∧ ∀𝑏 ∈ ω ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))))
303, 29mprg 3156 . . . 4 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎): 𝑎 ∈ ω (𝑅1𝑎)–1-1→ω
31 rdgfun 8046 . . . . . 6 Fun rec(𝐺, ∅)
32 funiunfv 7004 . . . . . . 7 (Fun rec(𝐺, ∅) → 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅) “ ω))
3332eqcomd 2831 . . . . . 6 (Fun rec(𝐺, ∅) → (rec(𝐺, ∅) “ ω) = 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎))
34 f1eq1 6566 . . . . . 6 ( (rec(𝐺, ∅) “ ω) = 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎) → ( (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1→ω ↔ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎): (𝑅1 “ ω)–1-1→ω))
3531, 33, 34mp2b 10 . . . . 5 ( (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1→ω ↔ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎): (𝑅1 “ ω)–1-1→ω)
36 r1funlim 9187 . . . . . . 7 (Fun 𝑅1 ∧ Lim dom 𝑅1)
3736simpli 484 . . . . . 6 Fun 𝑅1
38 funiunfv 7004 . . . . . 6 (Fun 𝑅1 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω))
39 f1eq2 6567 . . . . . 6 ( 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω) → ( 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎): 𝑎 ∈ ω (𝑅1𝑎)–1-1→ω ↔ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎): (𝑅1 “ ω)–1-1→ω))
4037, 38, 39mp2b 10 . . . . 5 ( 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎): 𝑎 ∈ ω (𝑅1𝑎)–1-1→ω ↔ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎): (𝑅1 “ ω)–1-1→ω)
4135, 40bitr4i 279 . . . 4 ( (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1→ω ↔ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎): 𝑎 ∈ ω (𝑅1𝑎)–1-1→ω)
4230, 41mpbir 232 . . 3 (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1→ω
43 rnuni 6004 . . . 4 ran (rec(𝐺, ∅) “ ω) = 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎
44 eliun 4920 . . . . . 6 (𝑏 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 ↔ ∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎)
45 df-rex 3148 . . . . . 6 (∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎 ↔ ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
46 funfn 6381 . . . . . . . . . . . 12 (Fun rec(𝐺, ∅) ↔ rec(𝐺, ∅) Fn dom rec(𝐺, ∅))
4731, 46mpbi 231 . . . . . . . . . . 11 rec(𝐺, ∅) Fn dom rec(𝐺, ∅)
48 rdgdmlim 8047 . . . . . . . . . . . 12 Lim dom rec(𝐺, ∅)
49 limomss 7576 . . . . . . . . . . . 12 (Lim dom rec(𝐺, ∅) → ω ⊆ dom rec(𝐺, ∅))
5048, 49ax-mp 5 . . . . . . . . . . 11 ω ⊆ dom rec(𝐺, ∅)
51 fvelimab 6733 . . . . . . . . . . 11 ((rec(𝐺, ∅) Fn dom rec(𝐺, ∅) ∧ ω ⊆ dom rec(𝐺, ∅)) → (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ ∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎))
5247, 50, 51mp2an 688 . . . . . . . . . 10 (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ ∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎)
534, 5ackbij2lem2 9654 . . . . . . . . . . . . . 14 (𝑐 ∈ ω → (rec(𝐺, ∅)‘𝑐):(𝑅1𝑐)–1-1-onto→(card‘(𝑅1𝑐)))
54 f1ofo 6618 . . . . . . . . . . . . . 14 ((rec(𝐺, ∅)‘𝑐):(𝑅1𝑐)–1-1-onto→(card‘(𝑅1𝑐)) → (rec(𝐺, ∅)‘𝑐):(𝑅1𝑐)–onto→(card‘(𝑅1𝑐)))
55 forn 6589 . . . . . . . . . . . . . 14 ((rec(𝐺, ∅)‘𝑐):(𝑅1𝑐)–onto→(card‘(𝑅1𝑐)) → ran (rec(𝐺, ∅)‘𝑐) = (card‘(𝑅1𝑐)))
5653, 54, 553syl 18 . . . . . . . . . . . . 13 (𝑐 ∈ ω → ran (rec(𝐺, ∅)‘𝑐) = (card‘(𝑅1𝑐)))
57 r1fin 9194 . . . . . . . . . . . . . . 15 (𝑐 ∈ ω → (𝑅1𝑐) ∈ Fin)
58 ficardom 9382 . . . . . . . . . . . . . . 15 ((𝑅1𝑐) ∈ Fin → (card‘(𝑅1𝑐)) ∈ ω)
5957, 58syl 17 . . . . . . . . . . . . . 14 (𝑐 ∈ ω → (card‘(𝑅1𝑐)) ∈ ω)
60 ordelss 6204 . . . . . . . . . . . . . 14 ((Ord ω ∧ (card‘(𝑅1𝑐)) ∈ ω) → (card‘(𝑅1𝑐)) ⊆ ω)
619, 59, 60sylancr 587 . . . . . . . . . . . . 13 (𝑐 ∈ ω → (card‘(𝑅1𝑐)) ⊆ ω)
6256, 61eqsstrd 4008 . . . . . . . . . . . 12 (𝑐 ∈ ω → ran (rec(𝐺, ∅)‘𝑐) ⊆ ω)
63 rneq 5804 . . . . . . . . . . . . 13 ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran (rec(𝐺, ∅)‘𝑐) = ran 𝑎)
6463sseq1d 4001 . . . . . . . . . . . 12 ((rec(𝐺, ∅)‘𝑐) = 𝑎 → (ran (rec(𝐺, ∅)‘𝑐) ⊆ ω ↔ ran 𝑎 ⊆ ω))
6562, 64syl5ibcom 246 . . . . . . . . . . 11 (𝑐 ∈ ω → ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran 𝑎 ⊆ ω))
6665rexlimiv 3284 . . . . . . . . . 10 (∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎 → ran 𝑎 ⊆ ω)
6752, 66sylbi 218 . . . . . . . . 9 (𝑎 ∈ (rec(𝐺, ∅) “ ω) → ran 𝑎 ⊆ ω)
6867sselda 3970 . . . . . . . 8 ((𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
6968exlimiv 1924 . . . . . . 7 (∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
70 peano2 7593 . . . . . . . . 9 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
71 fnfvima 6992 . . . . . . . . 9 ((rec(𝐺, ∅) Fn dom rec(𝐺, ∅) ∧ ω ⊆ dom rec(𝐺, ∅) ∧ suc 𝑏 ∈ ω) → (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω))
7247, 50, 70, 71mp3an12i 1458 . . . . . . . 8 (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω))
73 vex 3502 . . . . . . . . . 10 𝑏 ∈ V
74 cardnn 9384 . . . . . . . . . . . 12 (suc 𝑏 ∈ ω → (card‘suc 𝑏) = suc 𝑏)
75 fvex 6679 . . . . . . . . . . . . . 14 (𝑅1‘suc 𝑏) ∈ V
7636simpri 486 . . . . . . . . . . . . . . . . 17 Lim dom 𝑅1
77 limomss 7576 . . . . . . . . . . . . . . . . 17 (Lim dom 𝑅1 → ω ⊆ dom 𝑅1)
7876, 77ax-mp 5 . . . . . . . . . . . . . . . 16 ω ⊆ dom 𝑅1
7978sseli 3966 . . . . . . . . . . . . . . 15 (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom 𝑅1)
80 onssr1 9252 . . . . . . . . . . . . . . 15 (suc 𝑏 ∈ dom 𝑅1 → suc 𝑏 ⊆ (𝑅1‘suc 𝑏))
8179, 80syl 17 . . . . . . . . . . . . . 14 (suc 𝑏 ∈ ω → suc 𝑏 ⊆ (𝑅1‘suc 𝑏))
82 ssdomg 8547 . . . . . . . . . . . . . 14 ((𝑅1‘suc 𝑏) ∈ V → (suc 𝑏 ⊆ (𝑅1‘suc 𝑏) → suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
8375, 81, 82mpsyl 68 . . . . . . . . . . . . 13 (suc 𝑏 ∈ ω → suc 𝑏 ≼ (𝑅1‘suc 𝑏))
84 nnon 7577 . . . . . . . . . . . . . . 15 (suc 𝑏 ∈ ω → suc 𝑏 ∈ On)
85 onenon 9370 . . . . . . . . . . . . . . 15 (suc 𝑏 ∈ On → suc 𝑏 ∈ dom card)
8684, 85syl 17 . . . . . . . . . . . . . 14 (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom card)
87 r1fin 9194 . . . . . . . . . . . . . . 15 (suc 𝑏 ∈ ω → (𝑅1‘suc 𝑏) ∈ Fin)
88 finnum 9369 . . . . . . . . . . . . . . 15 ((𝑅1‘suc 𝑏) ∈ Fin → (𝑅1‘suc 𝑏) ∈ dom card)
8987, 88syl 17 . . . . . . . . . . . . . 14 (suc 𝑏 ∈ ω → (𝑅1‘suc 𝑏) ∈ dom card)
90 carddom2 9398 . . . . . . . . . . . . . 14 ((suc 𝑏 ∈ dom card ∧ (𝑅1‘suc 𝑏) ∈ dom card) → ((card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)) ↔ suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
9186, 89, 90syl2anc 584 . . . . . . . . . . . . 13 (suc 𝑏 ∈ ω → ((card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)) ↔ suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
9283, 91mpbird 258 . . . . . . . . . . . 12 (suc 𝑏 ∈ ω → (card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)))
9374, 92eqsstrrd 4009 . . . . . . . . . . 11 (suc 𝑏 ∈ ω → suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)))
9470, 93syl 17 . . . . . . . . . 10 (𝑏 ∈ ω → suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)))
95 sucssel 6280 . . . . . . . . . 10 (𝑏 ∈ V → (suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)) → 𝑏 ∈ (card‘(𝑅1‘suc 𝑏))))
9673, 94, 95mpsyl 68 . . . . . . . . 9 (𝑏 ∈ ω → 𝑏 ∈ (card‘(𝑅1‘suc 𝑏)))
974, 5ackbij2lem2 9654 . . . . . . . . . 10 (suc 𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)))
98 f1ofo 6618 . . . . . . . . . 10 ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)) → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–onto→(card‘(𝑅1‘suc 𝑏)))
99 forn 6589 . . . . . . . . . 10 ((rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–onto→(card‘(𝑅1‘suc 𝑏)) → ran (rec(𝐺, ∅)‘suc 𝑏) = (card‘(𝑅1‘suc 𝑏)))
10070, 97, 98, 994syl 19 . . . . . . . . 9 (𝑏 ∈ ω → ran (rec(𝐺, ∅)‘suc 𝑏) = (card‘(𝑅1‘suc 𝑏)))
10196, 100eleqtrrd 2920 . . . . . . . 8 (𝑏 ∈ ω → 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏))
102 fvex 6679 . . . . . . . . 9 (rec(𝐺, ∅)‘suc 𝑏) ∈ V
103 eleq1 2904 . . . . . . . . . 10 (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω)))
104 rneq 5804 . . . . . . . . . . 11 (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → ran 𝑎 = ran (rec(𝐺, ∅)‘suc 𝑏))
105104eleq2d 2902 . . . . . . . . . 10 (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑏 ∈ ran 𝑎𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏)))
106103, 105anbi12d 630 . . . . . . . . 9 (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → ((𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) ↔ ((rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏))))
107102, 106spcev 3610 . . . . . . . 8 (((rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏)) → ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
10872, 101, 107syl2anc 584 . . . . . . 7 (𝑏 ∈ ω → ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
10969, 108impbii 210 . . . . . 6 (∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) ↔ 𝑏 ∈ ω)
11044, 45, 1093bitri 298 . . . . 5 (𝑏 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎𝑏 ∈ ω)
111110eqriv 2822 . . . 4 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 = ω
11243, 111eqtri 2848 . . 3 ran (rec(𝐺, ∅) “ ω) = ω
113 dff1o5 6620 . . 3 ( (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1-onto→ω ↔ ( (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1→ω ∧ ran (rec(𝐺, ∅) “ ω) = ω))
11442, 112, 113mpbir2an 707 . 2 (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1-onto→ω
115 ackbij.h . . 3 𝐻 = (rec(𝐺, ∅) “ ω)
116 f1oeq1 6600 . . 3 (𝐻 = (rec(𝐺, ∅) “ ω) → (𝐻: (𝑅1 “ ω)–1-1-onto→ω ↔ (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1-onto→ω))
117115, 116ax-mp 5 . 2 (𝐻: (𝑅1 “ ω)–1-1-onto→ω ↔ (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1-onto→ω)
118114, 117mpbir 232 1 𝐻: (𝑅1 “ ω)–1-1-onto→ω
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wex 1773  wcel 2107  wral 3142  wrex 3143  Vcvv 3499  cin 3938  wss 3939  c0 4294  𝒫 cpw 4541  {csn 4563   cuni 4836   ciun 4916   class class class wbr 5062  cmpt 5142   × cxp 5551  dom cdm 5553  ran crn 5554  cima 5556  Ord word 6187  Oncon0 6188  Lim wlim 6189  suc csuc 6190  Fun wfun 6345   Fn wfn 6346  1-1wf1 6348  ontowfo 6349  1-1-ontowf1o 6350  cfv 6351  ωcom 7571  reccrdg 8039  cdom 8499  Fincfn 8501  𝑅1cr1 9183  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-r1 9185  df-rank 9186  df-dju 9322  df-card 9360
This theorem is referenced by:  r1om  9658
  Copyright terms: Public domain W3C validator