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Theorem dfac12lem3 10122
Description: Lemma for dfac12 10126. (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 6891 . . . 4 (𝐺𝐴) ∈ V
21rnex 7885 . . 3 ran (𝐺𝐴) ∈ V
3 ssid 4000 . . . . 5 𝐴𝐴
4 dfac12.1 . . . . . 6 (𝜑𝐴 ∈ On)
5 sseq1 4003 . . . . . . . . 9 (𝑚 = 𝑛 → (𝑚𝐴𝑛𝐴))
6 fveq2 6878 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝐺𝑚) = (𝐺𝑛))
7 f1eq1 6769 . . . . . . . . . . 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 6878 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑅1𝑚) = (𝑅1𝑛))
10 f1eq2 6770 . . . . . . . . . . 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 278 . . . . . . . . 9 (𝑚 = 𝑛 → ((𝐺𝑚):(𝑅1𝑚)–1-1→On ↔ (𝐺𝑛):(𝑅1𝑛)–1-1→On))
135, 12imbi12d 344 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On) ↔ (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)))
1413imbi2d 340 . . . . . . 7 (𝑚 = 𝑛 → ((𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On)) ↔ (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On))))
15 sseq1 4003 . . . . . . . . 9 (𝑚 = 𝐴 → (𝑚𝐴𝐴𝐴))
16 fveq2 6878 . . . . . . . . . . 11 (𝑚 = 𝐴 → (𝐺𝑚) = (𝐺𝐴))
17 f1eq1 6769 . . . . . . . . . . 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 6878 . . . . . . . . . . 11 (𝑚 = 𝐴 → (𝑅1𝑚) = (𝑅1𝐴))
20 f1eq2 6770 . . . . . . . . . . 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 278 . . . . . . . . 9 (𝑚 = 𝐴 → ((𝐺𝑚):(𝑅1𝑚)–1-1→On ↔ (𝐺𝐴):(𝑅1𝐴)–1-1→On))
2315, 22imbi12d 344 . . . . . . . 8 (𝑚 = 𝐴 → ((𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On) ↔ (𝐴𝐴 → (𝐺𝐴):(𝑅1𝐴)–1-1→On)))
2423imbi2d 340 . . . . . . 7 (𝑚 = 𝐴 → ((𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On)) ↔ (𝜑 → (𝐴𝐴 → (𝐺𝐴):(𝑅1𝐴)–1-1→On))))
25 r19.21v 3178 . . . . . . . 8 (∀𝑛𝑚 (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)) ↔ (𝜑 → ∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)))
26 eloni 6363 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ On → Ord 𝑚)
2726ad2antrl 726 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → Ord 𝑚)
28 ordelss 6369 . . . . . . . . . . . . . . . . 17 ((Ord 𝑚𝑛𝑚) → 𝑛𝑚)
2927, 28sylan 580 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑛𝑚)
30 simplrr 776 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑚𝐴)
3129, 30sstrd 3988 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑛𝐴)
32 pm5.5 361 . . . . . . . . . . . . . . 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 3174 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) ↔ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On))
354ad2antrr 724 . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝑚 ∈ On)
40 eqid 2731 . . . . . . . . . . . . . . 15 (OrdIso( E , ran (𝐺 𝑚)) ∘ (𝐺 𝑚)) = (OrdIso( E , ran (𝐺 𝑚)) ∘ (𝐺 𝑚))
41 simplrr 776 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝑚𝐴)
42 simpr 485 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On)
43 fveq2 6878 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝐺𝑛) = (𝐺𝑧))
44 f1eq1 6769 . . . . . . . . . . . . . . . . . . 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 6878 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝑅1𝑛) = (𝑅1𝑧))
47 f1eq2 6770 . . . . . . . . . . . . . . . . . . 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 278 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑧 → ((𝐺𝑛):(𝑅1𝑛)–1-1→On ↔ (𝐺𝑧):(𝑅1𝑧)–1-1→On))
5049cbvralvw 3233 . . . . . . . . . . . . . . . 16 (∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On ↔ ∀𝑧𝑚 (𝐺𝑧):(𝑅1𝑧)–1-1→On)
5142, 50sylib 217 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → ∀𝑧𝑚 (𝐺𝑧):(𝑅1𝑧)–1-1→On)
5235, 37, 38, 39, 40, 41, 51dfac12lem2 10121 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝐺𝑚):(𝑅1𝑚)–1-1→On)
5352ex 413 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On → (𝐺𝑚):(𝑅1𝑚)–1-1→On))
5434, 53sylbid 239 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝐺𝑚):(𝑅1𝑚)–1-1→On))
5554expr 457 . . . . . . . . . . 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 414 . . . . . . . . 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, 58biimtrid 241 . . . . . . 7 (𝑚 ∈ On → (∀𝑛𝑚 (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)) → (𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On))))
6014, 24, 59tfis3 7830 . . . . . 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 6774 . . . 4 ((𝐺𝐴):(𝑅1𝐴)–1-1→On → (𝐺𝐴):(𝑅1𝐴)⟶On)
64 frn 6711 . . . 4 ((𝐺𝐴):(𝑅1𝐴)⟶On → ran (𝐺𝐴) ⊆ On)
6562, 63, 643syl 18 . . 3 (𝜑 → ran (𝐺𝐴) ⊆ On)
66 onssnum 10017 . . 3 ((ran (𝐺𝐴) ∈ V ∧ ran (𝐺𝐴) ⊆ On) → ran (𝐺𝐴) ∈ dom card)
672, 65, 66sylancr 587 . 2 (𝜑 → ran (𝐺𝐴) ∈ dom card)
68 f1f1orn 6831 . . . 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 6891 . . . 4 (𝑅1𝐴) ∈ V
7170f1oen 8952 . . 3 ((𝐺𝐴):(𝑅1𝐴)–1-1-onto→ran (𝐺𝐴) → (𝑅1𝐴) ≈ ran (𝐺𝐴))
72 ennum 9924 . . 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 256 1 (𝜑 → (𝑅1𝐴) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3060  Vcvv 3473  wss 3944  ifcif 4522  𝒫 cpw 4596   cuni 4901   class class class wbr 5141  cmpt 5224   E cep 5572  ccnv 5668  dom cdm 5669  ran crn 5670  cima 5672  ccom 5673  Ord word 6352  Oncon0 6353  suc csuc 6355  wf 6528  1-1wf1 6529  1-1-ontowf1o 6531  cfv 6532  (class class class)co 7393  recscrecs 8352   +o coa 8445   ·o comu 8446  cen 8919  OrdIsocoi 9486  harchar 9533  𝑅1cr1 9739  rankcrnk 9740  cardccrd 9912
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-om 7839  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-oadd 8452  df-omul 8453  df-er 8686  df-en 8923  df-dom 8924  df-oi 9487  df-har 9534  df-r1 9741  df-rank 9742  df-card 9916
This theorem is referenced by:  dfac12r  10123
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