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Theorem dfac12lem3 9005
Description: Lemma for dfac12 9009. (Contributed by Mario Carneiro, 29-May-2015.)
Hypotheses
Ref Expression
dfac12.1 (𝜑𝐴 ∈ On)
dfac12.3 (𝜑𝐹:𝒫 (har‘(𝑅1𝐴))–1-1→On)
dfac12.4 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))
Assertion
Ref Expression
dfac12lem3 (𝜑 → (𝑅1𝐴) ∈ dom card)
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐺   𝜑,𝑦   𝑥,𝐹,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)

Proof of Theorem dfac12lem3
Dummy variables 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6239 . . . 4 (𝐺𝐴) ∈ V
21rnex 7142 . . 3 ran (𝐺𝐴) ∈ V
3 ssid 3657 . . . . 5 𝐴𝐴
4 dfac12.1 . . . . . 6 (𝜑𝐴 ∈ On)
5 sseq1 3659 . . . . . . . . 9 (𝑚 = 𝑛 → (𝑚𝐴𝑛𝐴))
6 fveq2 6229 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝐺𝑚) = (𝐺𝑛))
7 f1eq1 6134 . . . . . . . . . . 11 ((𝐺𝑚) = (𝐺𝑛) → ((𝐺𝑚):(𝑅1𝑚)–1-1→On ↔ (𝐺𝑛):(𝑅1𝑚)–1-1→On))
86, 7syl 17 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝐺𝑚):(𝑅1𝑚)–1-1→On ↔ (𝐺𝑛):(𝑅1𝑚)–1-1→On))
9 fveq2 6229 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑅1𝑚) = (𝑅1𝑛))
10 f1eq2 6135 . . . . . . . . . . 11 ((𝑅1𝑚) = (𝑅1𝑛) → ((𝐺𝑛):(𝑅1𝑚)–1-1→On ↔ (𝐺𝑛):(𝑅1𝑛)–1-1→On))
119, 10syl 17 . . . . . . . . . 10 (𝑚 = 𝑛 → ((𝐺𝑛):(𝑅1𝑚)–1-1→On ↔ (𝐺𝑛):(𝑅1𝑛)–1-1→On))
128, 11bitrd 268 . . . . . . . . 9 (𝑚 = 𝑛 → ((𝐺𝑚):(𝑅1𝑚)–1-1→On ↔ (𝐺𝑛):(𝑅1𝑛)–1-1→On))
135, 12imbi12d 333 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On) ↔ (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)))
1413imbi2d 329 . . . . . . 7 (𝑚 = 𝑛 → ((𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On)) ↔ (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On))))
15 sseq1 3659 . . . . . . . . 9 (𝑚 = 𝐴 → (𝑚𝐴𝐴𝐴))
16 fveq2 6229 . . . . . . . . . . 11 (𝑚 = 𝐴 → (𝐺𝑚) = (𝐺𝐴))
17 f1eq1 6134 . . . . . . . . . . 11 ((𝐺𝑚) = (𝐺𝐴) → ((𝐺𝑚):(𝑅1𝑚)–1-1→On ↔ (𝐺𝐴):(𝑅1𝑚)–1-1→On))
1816, 17syl 17 . . . . . . . . . 10 (𝑚 = 𝐴 → ((𝐺𝑚):(𝑅1𝑚)–1-1→On ↔ (𝐺𝐴):(𝑅1𝑚)–1-1→On))
19 fveq2 6229 . . . . . . . . . . 11 (𝑚 = 𝐴 → (𝑅1𝑚) = (𝑅1𝐴))
20 f1eq2 6135 . . . . . . . . . . 11 ((𝑅1𝑚) = (𝑅1𝐴) → ((𝐺𝐴):(𝑅1𝑚)–1-1→On ↔ (𝐺𝐴):(𝑅1𝐴)–1-1→On))
2119, 20syl 17 . . . . . . . . . 10 (𝑚 = 𝐴 → ((𝐺𝐴):(𝑅1𝑚)–1-1→On ↔ (𝐺𝐴):(𝑅1𝐴)–1-1→On))
2218, 21bitrd 268 . . . . . . . . 9 (𝑚 = 𝐴 → ((𝐺𝑚):(𝑅1𝑚)–1-1→On ↔ (𝐺𝐴):(𝑅1𝐴)–1-1→On))
2315, 22imbi12d 333 . . . . . . . 8 (𝑚 = 𝐴 → ((𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On) ↔ (𝐴𝐴 → (𝐺𝐴):(𝑅1𝐴)–1-1→On)))
2423imbi2d 329 . . . . . . 7 (𝑚 = 𝐴 → ((𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On)) ↔ (𝜑 → (𝐴𝐴 → (𝐺𝐴):(𝑅1𝐴)–1-1→On))))
25 r19.21v 2989 . . . . . . . 8 (∀𝑛𝑚 (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)) ↔ (𝜑 → ∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)))
26 eloni 5771 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ On → Ord 𝑚)
2726ad2antrl 764 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → Ord 𝑚)
28 ordelss 5777 . . . . . . . . . . . . . . . . 17 ((Ord 𝑚𝑛𝑚) → 𝑛𝑚)
2927, 28sylan 487 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑛𝑚)
30 simplrr 818 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑚𝐴)
3129, 30sstrd 3646 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑛𝐴)
32 pm5.5 350 . . . . . . . . . . . . . . 15 (𝑛𝐴 → ((𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) ↔ (𝐺𝑛):(𝑅1𝑛)–1-1→On))
3331, 32syl 17 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → ((𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) ↔ (𝐺𝑛):(𝑅1𝑛)–1-1→On))
3433ralbidva 3014 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) ↔ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On))
354ad2antrr 762 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝐴 ∈ On)
36 dfac12.3 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝒫 (har‘(𝑅1𝐴))–1-1→On)
3736ad2antrr 762 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝐹:𝒫 (har‘(𝑅1𝐴))–1-1→On)
38 dfac12.4 . . . . . . . . . . . . . . 15 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = dom 𝑥, ((suc ran ran 𝑥 ·𝑜 (rank‘𝑦)) +𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((OrdIso( E , ran (𝑥 dom 𝑥)) ∘ (𝑥 dom 𝑥)) “ 𝑦))))))
39 simplrl 817 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝑚 ∈ On)
40 eqid 2651 . . . . . . . . . . . . . . 15 (OrdIso( E , ran (𝐺 𝑚)) ∘ (𝐺 𝑚)) = (OrdIso( E , ran (𝐺 𝑚)) ∘ (𝐺 𝑚))
41 simplrr 818 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝑚𝐴)
42 simpr 476 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On)
43 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝐺𝑛) = (𝐺𝑧))
44 f1eq1 6134 . . . . . . . . . . . . . . . . . . 19 ((𝐺𝑛) = (𝐺𝑧) → ((𝐺𝑛):(𝑅1𝑛)–1-1→On ↔ (𝐺𝑧):(𝑅1𝑛)–1-1→On))
4543, 44syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑧 → ((𝐺𝑛):(𝑅1𝑛)–1-1→On ↔ (𝐺𝑧):(𝑅1𝑛)–1-1→On))
46 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝑅1𝑛) = (𝑅1𝑧))
47 f1eq2 6135 . . . . . . . . . . . . . . . . . . 19 ((𝑅1𝑛) = (𝑅1𝑧) → ((𝐺𝑧):(𝑅1𝑛)–1-1→On ↔ (𝐺𝑧):(𝑅1𝑧)–1-1→On))
4846, 47syl 17 . . . . . . . . . . . . . . . . . 18 (𝑛 = 𝑧 → ((𝐺𝑧):(𝑅1𝑛)–1-1→On ↔ (𝐺𝑧):(𝑅1𝑧)–1-1→On))
4945, 48bitrd 268 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑧 → ((𝐺𝑛):(𝑅1𝑛)–1-1→On ↔ (𝐺𝑧):(𝑅1𝑧)–1-1→On))
5049cbvralv 3201 . . . . . . . . . . . . . . . 16 (∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On ↔ ∀𝑧𝑚 (𝐺𝑧):(𝑅1𝑧)–1-1→On)
5142, 50sylib 208 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → ∀𝑧𝑚 (𝐺𝑧):(𝑅1𝑧)–1-1→On)
5235, 37, 38, 39, 40, 41, 51dfac12lem2 9004 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝐺𝑚):(𝑅1𝑚)–1-1→On)
5352ex 449 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On → (𝐺𝑚):(𝑅1𝑚)–1-1→On))
5434, 53sylbid 230 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝐺𝑚):(𝑅1𝑚)–1-1→On))
5554expr 642 . . . . . . . . . . 11 ((𝜑𝑚 ∈ On) → (𝑚𝐴 → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝐺𝑚):(𝑅1𝑚)–1-1→On)))
5655com23 86 . . . . . . . . . 10 ((𝜑𝑚 ∈ On) → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On)))
5756expcom 450 . . . . . . . . 9 (𝑚 ∈ On → (𝜑 → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On))))
5857a2d 29 . . . . . . . 8 (𝑚 ∈ On → ((𝜑 → ∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)) → (𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On))))
5925, 58syl5bi 232 . . . . . . 7 (𝑚 ∈ On → (∀𝑛𝑚 (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)) → (𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On))))
6014, 24, 59tfis3 7099 . . . . . 6 (𝐴 ∈ On → (𝜑 → (𝐴𝐴 → (𝐺𝐴):(𝑅1𝐴)–1-1→On)))
614, 60mpcom 38 . . . . 5 (𝜑 → (𝐴𝐴 → (𝐺𝐴):(𝑅1𝐴)–1-1→On))
623, 61mpi 20 . . . 4 (𝜑 → (𝐺𝐴):(𝑅1𝐴)–1-1→On)
63 f1f 6139 . . . 4 ((𝐺𝐴):(𝑅1𝐴)–1-1→On → (𝐺𝐴):(𝑅1𝐴)⟶On)
64 frn 6091 . . . 4 ((𝐺𝐴):(𝑅1𝐴)⟶On → ran (𝐺𝐴) ⊆ On)
6562, 63, 643syl 18 . . 3 (𝜑 → ran (𝐺𝐴) ⊆ On)
66 onssnum 8901 . . 3 ((ran (𝐺𝐴) ∈ V ∧ ran (𝐺𝐴) ⊆ On) → ran (𝐺𝐴) ∈ dom card)
672, 65, 66sylancr 696 . 2 (𝜑 → ran (𝐺𝐴) ∈ dom card)
68 f1f1orn 6186 . . . 4 ((𝐺𝐴):(𝑅1𝐴)–1-1→On → (𝐺𝐴):(𝑅1𝐴)–1-1-onto→ran (𝐺𝐴))
6962, 68syl 17 . . 3 (𝜑 → (𝐺𝐴):(𝑅1𝐴)–1-1-onto→ran (𝐺𝐴))
70 fvex 6239 . . . 4 (𝑅1𝐴) ∈ V
7170f1oen 8018 . . 3 ((𝐺𝐴):(𝑅1𝐴)–1-1-onto→ran (𝐺𝐴) → (𝑅1𝐴) ≈ ran (𝐺𝐴))
72 ennum 8811 . . 3 ((𝑅1𝐴) ≈ ran (𝐺𝐴) → ((𝑅1𝐴) ∈ dom card ↔ ran (𝐺𝐴) ∈ dom card))
7369, 71, 723syl 18 . 2 (𝜑 → ((𝑅1𝐴) ∈ dom card ↔ ran (𝐺𝐴) ∈ dom card))
7467, 73mpbird 247 1 (𝜑 → (𝑅1𝐴) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  wss 3607  ifcif 4119  𝒫 cpw 4191   cuni 4468   class class class wbr 4685  cmpt 4762   E cep 5057  ccnv 5142  dom cdm 5143  ran crn 5144  cima 5146  ccom 5147  Ord word 5760  Oncon0 5761  suc csuc 5763  wf 5922  1-1wf1 5923  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  recscrecs 7512   +𝑜 coa 7602   ·𝑜 comu 7603  cen 7994  OrdIsocoi 8455  harchar 8502  𝑅1cr1 8663  rankcrnk 8664  cardccrd 8799
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
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  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-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  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-se 5103  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-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-omul 7610  df-er 7787  df-en 7998  df-dom 7999  df-oi 8456  df-har 8504  df-r1 8665  df-rank 8666  df-card 8803
This theorem is referenced by:  dfac12r  9006
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