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Theorem dfac12lem3 9250
Description: Lemma for dfac12 9254. (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 6419 . . . 4 (𝐺𝐴) ∈ V
21rnex 7328 . . 3 ran (𝐺𝐴) ∈ V
3 ssid 3818 . . . . 5 𝐴𝐴
4 dfac12.1 . . . . . 6 (𝜑𝐴 ∈ On)
5 sseq1 3821 . . . . . . . . 9 (𝑚 = 𝑛 → (𝑚𝐴𝑛𝐴))
6 fveq2 6406 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝐺𝑚) = (𝐺𝑛))
7 f1eq1 6309 . . . . . . . . . . 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 6406 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑅1𝑚) = (𝑅1𝑛))
10 f1eq2 6310 . . . . . . . . . . 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 270 . . . . . . . . 9 (𝑚 = 𝑛 → ((𝐺𝑚):(𝑅1𝑚)–1-1→On ↔ (𝐺𝑛):(𝑅1𝑛)–1-1→On))
135, 12imbi12d 335 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On) ↔ (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)))
1413imbi2d 331 . . . . . . 7 (𝑚 = 𝑛 → ((𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On)) ↔ (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On))))
15 sseq1 3821 . . . . . . . . 9 (𝑚 = 𝐴 → (𝑚𝐴𝐴𝐴))
16 fveq2 6406 . . . . . . . . . . 11 (𝑚 = 𝐴 → (𝐺𝑚) = (𝐺𝐴))
17 f1eq1 6309 . . . . . . . . . . 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 6406 . . . . . . . . . . 11 (𝑚 = 𝐴 → (𝑅1𝑚) = (𝑅1𝐴))
20 f1eq2 6310 . . . . . . . . . . 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 270 . . . . . . . . 9 (𝑚 = 𝐴 → ((𝐺𝑚):(𝑅1𝑚)–1-1→On ↔ (𝐺𝐴):(𝑅1𝐴)–1-1→On))
2315, 22imbi12d 335 . . . . . . . 8 (𝑚 = 𝐴 → ((𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On) ↔ (𝐴𝐴 → (𝐺𝐴):(𝑅1𝐴)–1-1→On)))
2423imbi2d 331 . . . . . . 7 (𝑚 = 𝐴 → ((𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On)) ↔ (𝜑 → (𝐴𝐴 → (𝐺𝐴):(𝑅1𝐴)–1-1→On))))
25 r19.21v 3146 . . . . . . . 8 (∀𝑛𝑚 (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)) ↔ (𝜑 → ∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)))
26 eloni 5944 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ On → Ord 𝑚)
2726ad2antrl 710 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → Ord 𝑚)
28 ordelss 5950 . . . . . . . . . . . . . . . . 17 ((Ord 𝑚𝑛𝑚) → 𝑛𝑚)
2927, 28sylan 571 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑛𝑚)
30 simplrr 787 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑚𝐴)
3129, 30sstrd 3806 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑛𝐴)
32 pm5.5 352 . . . . . . . . . . . . . . 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 3171 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) ↔ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On))
354ad2antrr 708 . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . 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 786 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝑚 ∈ On)
40 eqid 2804 . . . . . . . . . . . . . . 15 (OrdIso( E , ran (𝐺 𝑚)) ∘ (𝐺 𝑚)) = (OrdIso( E , ran (𝐺 𝑚)) ∘ (𝐺 𝑚))
41 simplrr 787 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝑚𝐴)
42 simpr 473 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On)
43 fveq2 6406 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝐺𝑛) = (𝐺𝑧))
44 f1eq1 6309 . . . . . . . . . . . . . . . . . . 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 6406 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝑅1𝑛) = (𝑅1𝑧))
47 f1eq2 6310 . . . . . . . . . . . . . . . . . . 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 270 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑧 → ((𝐺𝑛):(𝑅1𝑛)–1-1→On ↔ (𝐺𝑧):(𝑅1𝑧)–1-1→On))
5049cbvralv 3358 . . . . . . . . . . . . . . . 16 (∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On ↔ ∀𝑧𝑚 (𝐺𝑧):(𝑅1𝑧)–1-1→On)
5142, 50sylib 209 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → ∀𝑧𝑚 (𝐺𝑧):(𝑅1𝑧)–1-1→On)
5235, 37, 38, 39, 40, 41, 51dfac12lem2 9249 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝐺𝑚):(𝑅1𝑚)–1-1→On)
5352ex 399 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On → (𝐺𝑚):(𝑅1𝑚)–1-1→On))
5434, 53sylbid 231 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝐺𝑚):(𝑅1𝑚)–1-1→On))
5554expr 446 . . . . . . . . . . 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 400 . . . . . . . . 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 233 . . . . . . 7 (𝑚 ∈ On → (∀𝑛𝑚 (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)) → (𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On))))
6014, 24, 59tfis3 7285 . . . . . 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 6314 . . . 4 ((𝐺𝐴):(𝑅1𝐴)–1-1→On → (𝐺𝐴):(𝑅1𝐴)⟶On)
64 frn 6260 . . . 4 ((𝐺𝐴):(𝑅1𝐴)⟶On → ran (𝐺𝐴) ⊆ On)
6562, 63, 643syl 18 . . 3 (𝜑 → ran (𝐺𝐴) ⊆ On)
66 onssnum 9144 . . 3 ((ran (𝐺𝐴) ∈ V ∧ ran (𝐺𝐴) ⊆ On) → ran (𝐺𝐴) ∈ dom card)
672, 65, 66sylancr 577 . 2 (𝜑 → ran (𝐺𝐴) ∈ dom card)
68 f1f1orn 6362 . . . 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 6419 . . . 4 (𝑅1𝐴) ∈ V
7170f1oen 8211 . . 3 ((𝐺𝐴):(𝑅1𝐴)–1-1-onto→ran (𝐺𝐴) → (𝑅1𝐴) ≈ ran (𝐺𝐴))
72 ennum 9054 . . 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 248 1 (𝜑 → (𝑅1𝐴) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  wral 3094  Vcvv 3389  wss 3767  ifcif 4277  𝒫 cpw 4349   cuni 4628   class class class wbr 4842  cmpt 4921   E cep 5221  ccnv 5308  dom cdm 5309  ran crn 5310  cima 5312  ccom 5313  Ord word 5933  Oncon0 5934  suc csuc 5936  wf 6095  1-1wf1 6096  1-1-ontowf1o 6098  cfv 6099  (class class class)co 6872  recscrecs 7701   +𝑜 coa 7791   ·𝑜 comu 7792  cen 8187  OrdIsocoi 8651  harchar 8698  𝑅1cr1 8870  rankcrnk 8871  cardccrd 9042
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5094  ax-un 7177
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ne 2977  df-ral 3099  df-rex 3100  df-reu 3101  df-rmo 3102  df-rab 3103  df-v 3391  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4115  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4843  df-opab 4905  df-mpt 4922  df-tr 4945  df-id 5217  df-eprel 5222  df-po 5230  df-so 5231  df-fr 5268  df-se 5269  df-we 5270  df-xp 5315  df-rel 5316  df-cnv 5317  df-co 5318  df-dm 5319  df-rn 5320  df-res 5321  df-ima 5322  df-pred 5891  df-ord 5937  df-on 5938  df-lim 5939  df-suc 5940  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-isom 6108  df-riota 6833  df-ov 6875  df-oprab 6876  df-mpt2 6877  df-om 7294  df-wrecs 7640  df-recs 7702  df-rdg 7740  df-oadd 7798  df-omul 7799  df-er 7977  df-en 8191  df-dom 8192  df-oi 8652  df-har 8700  df-r1 8872  df-rank 8873  df-card 9046
This theorem is referenced by:  dfac12r  9251
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