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Theorem ackbij2 9716
 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 6663 . . . . . 6 (𝑎 = 𝑏 → (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅)‘𝑏))
2 fvex 6676 . . . . . 6 (rec(𝐺, ∅)‘𝑎) ∈ V
31, 2f1iun 7655 . . . . 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 9713 . . . . . . . 8 (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1-onto→(card‘(𝑅1𝑎)))
7 f1of1 6606 . . . . . . . 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 7594 . . . . . . . 8 Ord ω
10 r1fin 9248 . . . . . . . . 9 (𝑎 ∈ ω → (𝑅1𝑎) ∈ Fin)
11 ficardom 9436 . . . . . . . . 9 ((𝑅1𝑎) ∈ Fin → (card‘(𝑅1𝑎)) ∈ ω)
1210, 11syl 17 . . . . . . . 8 (𝑎 ∈ ω → (card‘(𝑅1𝑎)) ∈ ω)
13 ordelss 6190 . . . . . . . 8 ((Ord ω ∧ (card‘(𝑅1𝑎)) ∈ ω) → (card‘(𝑅1𝑎)) ⊆ ω)
149, 12, 13sylancr 590 . . . . . . 7 (𝑎 ∈ ω → (card‘(𝑅1𝑎)) ⊆ ω)
15 f1ss 6571 . . . . . . 7 (((rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1→(card‘(𝑅1𝑎)) ∧ (card‘(𝑅1𝑎)) ⊆ ω) → (rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1→ω)
168, 14, 15syl2anc 587 . . . . . 6 (𝑎 ∈ ω → (rec(𝐺, ∅)‘𝑎):(𝑅1𝑎)–1-1→ω)
17 nnord 7593 . . . . . . . . 9 (𝑎 ∈ ω → Ord 𝑎)
18 nnord 7593 . . . . . . . . 9 (𝑏 ∈ ω → Ord 𝑏)
19 ordtri2or2 6270 . . . . . . . . 9 ((Ord 𝑎 ∧ Ord 𝑏) → (𝑎𝑏𝑏𝑎))
2017, 18, 19syl2an 598 . . . . . . . 8 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎𝑏𝑏𝑎))
214, 5ackbij2lem4 9715 . . . . . . . . . . 11 (((𝑏 ∈ ω ∧ 𝑎 ∈ ω) ∧ 𝑎𝑏) → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏))
2221ex 416 . . . . . . . . . 10 ((𝑏 ∈ ω ∧ 𝑎 ∈ ω) → (𝑎𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
2322ancoms 462 . . . . . . . . 9 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑎𝑏 → (rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏)))
244, 5ackbij2lem4 9715 . . . . . . . . . 10 (((𝑎 ∈ ω ∧ 𝑏 ∈ ω) ∧ 𝑏𝑎) → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎))
2524ex 416 . . . . . . . . 9 ((𝑎 ∈ ω ∧ 𝑏 ∈ ω) → (𝑏𝑎 → (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
2623, 25orim12d 962 . . . . . . . 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 3113 . . . . . 6 (𝑎 ∈ ω → ∀𝑏 ∈ ω ((rec(𝐺, ∅)‘𝑎) ⊆ (rec(𝐺, ∅)‘𝑏) ∨ (rec(𝐺, ∅)‘𝑏) ⊆ (rec(𝐺, ∅)‘𝑎)))
2916, 28jca 515 . . . . 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 3084 . . . 4 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎): 𝑎 ∈ ω (𝑅1𝑎)–1-1→ω
31 rdgfun 8068 . . . . . 6 Fun rec(𝐺, ∅)
32 funiunfv 7005 . . . . . . 7 (Fun rec(𝐺, ∅) → 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎) = (rec(𝐺, ∅) “ ω))
3332eqcomd 2764 . . . . . 6 (Fun rec(𝐺, ∅) → (rec(𝐺, ∅) “ ω) = 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎))
34 f1eq1 6560 . . . . . 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 9241 . . . . . . 7 (Fun 𝑅1 ∧ Lim dom 𝑅1)
3736simpli 487 . . . . . 6 Fun 𝑅1
38 funiunfv 7005 . . . . . 6 (Fun 𝑅1 𝑎 ∈ ω (𝑅1𝑎) = (𝑅1 “ ω))
39 f1eq2 6561 . . . . . 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 281 . . . 4 ( (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1→ω ↔ 𝑎 ∈ ω (rec(𝐺, ∅)‘𝑎): 𝑎 ∈ ω (𝑅1𝑎)–1-1→ω)
4230, 41mpbir 234 . . 3 (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1→ω
43 rnuni 5984 . . . 4 ran (rec(𝐺, ∅) “ ω) = 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎
44 eliun 4890 . . . . . 6 (𝑏 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 ↔ ∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎)
45 df-rex 3076 . . . . . 6 (∃𝑎 ∈ (rec(𝐺, ∅) “ ω)𝑏 ∈ ran 𝑎 ↔ ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
46 funfn 6370 . . . . . . . . . . . 12 (Fun rec(𝐺, ∅) ↔ rec(𝐺, ∅) Fn dom rec(𝐺, ∅))
4731, 46mpbi 233 . . . . . . . . . . 11 rec(𝐺, ∅) Fn dom rec(𝐺, ∅)
48 rdgdmlim 8069 . . . . . . . . . . . 12 Lim dom rec(𝐺, ∅)
49 limomss 7590 . . . . . . . . . . . 12 (Lim dom rec(𝐺, ∅) → ω ⊆ dom rec(𝐺, ∅))
5048, 49ax-mp 5 . . . . . . . . . . 11 ω ⊆ dom rec(𝐺, ∅)
51 fvelimab 6730 . . . . . . . . . . 11 ((rec(𝐺, ∅) Fn dom rec(𝐺, ∅) ∧ ω ⊆ dom rec(𝐺, ∅)) → (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ ∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎))
5247, 50, 51mp2an 691 . . . . . . . . . 10 (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ ∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎)
534, 5ackbij2lem2 9713 . . . . . . . . . . . . . 14 (𝑐 ∈ ω → (rec(𝐺, ∅)‘𝑐):(𝑅1𝑐)–1-1-onto→(card‘(𝑅1𝑐)))
54 f1ofo 6614 . . . . . . . . . . . . . 14 ((rec(𝐺, ∅)‘𝑐):(𝑅1𝑐)–1-1-onto→(card‘(𝑅1𝑐)) → (rec(𝐺, ∅)‘𝑐):(𝑅1𝑐)–onto→(card‘(𝑅1𝑐)))
55 forn 6584 . . . . . . . . . . . . . 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 9248 . . . . . . . . . . . . . . 15 (𝑐 ∈ ω → (𝑅1𝑐) ∈ Fin)
58 ficardom 9436 . . . . . . . . . . . . . . 15 ((𝑅1𝑐) ∈ Fin → (card‘(𝑅1𝑐)) ∈ ω)
5957, 58syl 17 . . . . . . . . . . . . . 14 (𝑐 ∈ ω → (card‘(𝑅1𝑐)) ∈ ω)
60 ordelss 6190 . . . . . . . . . . . . . 14 ((Ord ω ∧ (card‘(𝑅1𝑐)) ∈ ω) → (card‘(𝑅1𝑐)) ⊆ ω)
619, 59, 60sylancr 590 . . . . . . . . . . . . 13 (𝑐 ∈ ω → (card‘(𝑅1𝑐)) ⊆ ω)
6256, 61eqsstrd 3932 . . . . . . . . . . . 12 (𝑐 ∈ ω → ran (rec(𝐺, ∅)‘𝑐) ⊆ ω)
63 rneq 5782 . . . . . . . . . . . . 13 ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran (rec(𝐺, ∅)‘𝑐) = ran 𝑎)
6463sseq1d 3925 . . . . . . . . . . . 12 ((rec(𝐺, ∅)‘𝑐) = 𝑎 → (ran (rec(𝐺, ∅)‘𝑐) ⊆ ω ↔ ran 𝑎 ⊆ ω))
6562, 64syl5ibcom 248 . . . . . . . . . . 11 (𝑐 ∈ ω → ((rec(𝐺, ∅)‘𝑐) = 𝑎 → ran 𝑎 ⊆ ω))
6665rexlimiv 3204 . . . . . . . . . 10 (∃𝑐 ∈ ω (rec(𝐺, ∅)‘𝑐) = 𝑎 → ran 𝑎 ⊆ ω)
6752, 66sylbi 220 . . . . . . . . 9 (𝑎 ∈ (rec(𝐺, ∅) “ ω) → ran 𝑎 ⊆ ω)
6867sselda 3894 . . . . . . . 8 ((𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
6968exlimiv 1931 . . . . . . 7 (∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) → 𝑏 ∈ ω)
70 peano2 7607 . . . . . . . . 9 (𝑏 ∈ ω → suc 𝑏 ∈ ω)
71 fnfvima 6993 . . . . . . . . 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 1462 . . . . . . . 8 (𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω))
73 vex 3413 . . . . . . . . . 10 𝑏 ∈ V
74 cardnn 9438 . . . . . . . . . . . 12 (suc 𝑏 ∈ ω → (card‘suc 𝑏) = suc 𝑏)
75 fvex 6676 . . . . . . . . . . . . . 14 (𝑅1‘suc 𝑏) ∈ V
7636simpri 489 . . . . . . . . . . . . . . . . 17 Lim dom 𝑅1
77 limomss 7590 . . . . . . . . . . . . . . . . 17 (Lim dom 𝑅1 → ω ⊆ dom 𝑅1)
7876, 77ax-mp 5 . . . . . . . . . . . . . . . 16 ω ⊆ dom 𝑅1
7978sseli 3890 . . . . . . . . . . . . . . 15 (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom 𝑅1)
80 onssr1 9306 . . . . . . . . . . . . . . 15 (suc 𝑏 ∈ dom 𝑅1 → suc 𝑏 ⊆ (𝑅1‘suc 𝑏))
8179, 80syl 17 . . . . . . . . . . . . . 14 (suc 𝑏 ∈ ω → suc 𝑏 ⊆ (𝑅1‘suc 𝑏))
82 ssdomg 8586 . . . . . . . . . . . . . 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 7591 . . . . . . . . . . . . . . 15 (suc 𝑏 ∈ ω → suc 𝑏 ∈ On)
85 onenon 9424 . . . . . . . . . . . . . . 15 (suc 𝑏 ∈ On → suc 𝑏 ∈ dom card)
8684, 85syl 17 . . . . . . . . . . . . . 14 (suc 𝑏 ∈ ω → suc 𝑏 ∈ dom card)
87 r1fin 9248 . . . . . . . . . . . . . . 15 (suc 𝑏 ∈ ω → (𝑅1‘suc 𝑏) ∈ Fin)
88 finnum 9423 . . . . . . . . . . . . . . 15 ((𝑅1‘suc 𝑏) ∈ Fin → (𝑅1‘suc 𝑏) ∈ dom card)
8987, 88syl 17 . . . . . . . . . . . . . 14 (suc 𝑏 ∈ ω → (𝑅1‘suc 𝑏) ∈ dom card)
90 carddom2 9452 . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . 13 (suc 𝑏 ∈ ω → ((card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)) ↔ suc 𝑏 ≼ (𝑅1‘suc 𝑏)))
9283, 91mpbird 260 . . . . . . . . . . . 12 (suc 𝑏 ∈ ω → (card‘suc 𝑏) ⊆ (card‘(𝑅1‘suc 𝑏)))
9374, 92eqsstrrd 3933 . . . . . . . . . . 11 (suc 𝑏 ∈ ω → suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)))
9470, 93syl 17 . . . . . . . . . 10 (𝑏 ∈ ω → suc 𝑏 ⊆ (card‘(𝑅1‘suc 𝑏)))
95 sucssel 6266 . . . . . . . . . 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 9713 . . . . . . . . . 10 (suc 𝑏 ∈ ω → (rec(𝐺, ∅)‘suc 𝑏):(𝑅1‘suc 𝑏)–1-1-onto→(card‘(𝑅1‘suc 𝑏)))
98 f1ofo 6614 . . . . . . . . . 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 6584 . . . . . . . . . 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 2855 . . . . . . . 8 (𝑏 ∈ ω → 𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏))
102 fvex 6676 . . . . . . . . 9 (rec(𝐺, ∅)‘suc 𝑏) ∈ V
103 eleq1 2839 . . . . . . . . . 10 (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑎 ∈ (rec(𝐺, ∅) “ ω) ↔ (rec(𝐺, ∅)‘suc 𝑏) ∈ (rec(𝐺, ∅) “ ω)))
104 rneq 5782 . . . . . . . . . . 11 (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → ran 𝑎 = ran (rec(𝐺, ∅)‘suc 𝑏))
105104eleq2d 2837 . . . . . . . . . 10 (𝑎 = (rec(𝐺, ∅)‘suc 𝑏) → (𝑏 ∈ ran 𝑎𝑏 ∈ ran (rec(𝐺, ∅)‘suc 𝑏)))
106103, 105anbi12d 633 . . . . . . . . 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 3527 . . . . . . . 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 587 . . . . . . 7 (𝑏 ∈ ω → ∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎))
10969, 108impbii 212 . . . . . 6 (∃𝑎(𝑎 ∈ (rec(𝐺, ∅) “ ω) ∧ 𝑏 ∈ ran 𝑎) ↔ 𝑏 ∈ ω)
11044, 45, 1093bitri 300 . . . . 5 (𝑏 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎𝑏 ∈ ω)
111110eqriv 2755 . . . 4 𝑎 ∈ (rec(𝐺, ∅) “ ω)ran 𝑎 = ω
11243, 111eqtri 2781 . . 3 ran (rec(𝐺, ∅) “ ω) = ω
113 dff1o5 6616 . . 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 710 . 2 (rec(𝐺, ∅) “ ω): (𝑅1 “ ω)–1-1-onto→ω
115 ackbij.h . . 3 𝐻 = (rec(𝐺, ∅) “ ω)
116 f1oeq1 6595 . . 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 234 1 𝐻: (𝑅1 “ ω)–1-1-onto→ω
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  𝒫 cpw 4497  {csn 4525  ∪ cuni 4801  ∪ ciun 4886   class class class wbr 5036   ↦ cmpt 5116   × cxp 5526  dom cdm 5528  ran crn 5529   “ cima 5531  Ord word 6173  Oncon0 6174  Lim wlim 6175  suc csuc 6176  Fun wfun 6334   Fn wfn 6335  –1-1→wf1 6337  –onto→wfo 6338  –1-1-onto→wf1o 6339  ‘cfv 6340  ωcom 7585  reccrdg 8061   ≼ cdom 8538  Fincfn 8540  𝑅1cr1 9237  cardccrd 9410 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7586  df-1st 7699  df-2nd 7700  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-2o 8119  df-oadd 8122  df-er 8305  df-map 8424  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-r1 9239  df-rank 9240  df-dju 9376  df-card 9414 This theorem is referenced by:  r1om  9717
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