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Theorem dfac12lem3 10183
Description: Lemma for dfac12 10187. (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 6919 . . . 4 (𝐺𝐴) ∈ V
21rnex 7932 . . 3 ran (𝐺𝐴) ∈ V
3 ssid 4017 . . . . 5 𝐴𝐴
4 dfac12.1 . . . . . 6 (𝜑𝐴 ∈ On)
5 sseq1 4020 . . . . . . . . 9 (𝑚 = 𝑛 → (𝑚𝐴𝑛𝐴))
6 fveq2 6906 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝐺𝑚) = (𝐺𝑛))
7 f1eq1 6799 . . . . . . . . . . 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 6906 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑅1𝑚) = (𝑅1𝑛))
10 f1eq2 6800 . . . . . . . . . . 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 279 . . . . . . . . 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 4020 . . . . . . . . 9 (𝑚 = 𝐴 → (𝑚𝐴𝐴𝐴))
16 fveq2 6906 . . . . . . . . . . 11 (𝑚 = 𝐴 → (𝐺𝑚) = (𝐺𝐴))
17 f1eq1 6799 . . . . . . . . . . 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 6906 . . . . . . . . . . 11 (𝑚 = 𝐴 → (𝑅1𝑚) = (𝑅1𝐴))
20 f1eq2 6800 . . . . . . . . . . 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 279 . . . . . . . . 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 3177 . . . . . . . 8 (∀𝑛𝑚 (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)) ↔ (𝜑 → ∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)))
26 eloni 6395 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ On → Ord 𝑚)
2726ad2antrl 728 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → Ord 𝑚)
28 ordelss 6401 . . . . . . . . . . . . . . . . 17 ((Ord 𝑚𝑛𝑚) → 𝑛𝑚)
2927, 28sylan 580 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑛𝑚)
30 simplrr 778 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑚𝐴)
3129, 30sstrd 4005 . . . . . . . . . . . . . . 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 3173 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) ↔ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On))
354ad2antrr 726 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝑚 ∈ On)
40 eqid 2734 . . . . . . . . . . . . . . 15 (OrdIso( E , ran (𝐺 𝑚)) ∘ (𝐺 𝑚)) = (OrdIso( E , ran (𝐺 𝑚)) ∘ (𝐺 𝑚))
41 simplrr 778 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝑚𝐴)
42 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On)
43 fveq2 6906 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝐺𝑛) = (𝐺𝑧))
44 f1eq1 6799 . . . . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝑅1𝑛) = (𝑅1𝑧))
47 f1eq2 6800 . . . . . . . . . . . . . . . . . . 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 279 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑧 → ((𝐺𝑛):(𝑅1𝑛)–1-1→On ↔ (𝐺𝑧):(𝑅1𝑧)–1-1→On))
5049cbvralvw 3234 . . . . . . . . . . . . . . . 16 (∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On ↔ ∀𝑧𝑚 (𝐺𝑧):(𝑅1𝑧)–1-1→On)
5142, 50sylib 218 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → ∀𝑧𝑚 (𝐺𝑧):(𝑅1𝑧)–1-1→On)
5235, 37, 38, 39, 40, 41, 51dfac12lem2 10182 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝐺𝑚):(𝑅1𝑚)–1-1→On)
5352ex 412 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On → (𝐺𝑚):(𝑅1𝑚)–1-1→On))
5434, 53sylbid 240 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝐺𝑚):(𝑅1𝑚)–1-1→On))
5554expr 456 . . . . . . . . . . 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 413 . . . . . . . . 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 242 . . . . . . 7 (𝑚 ∈ On → (∀𝑛𝑚 (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)) → (𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On))))
6014, 24, 59tfis3 7878 . . . . . 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 6804 . . . 4 ((𝐺𝐴):(𝑅1𝐴)–1-1→On → (𝐺𝐴):(𝑅1𝐴)⟶On)
64 frn 6743 . . . 4 ((𝐺𝐴):(𝑅1𝐴)⟶On → ran (𝐺𝐴) ⊆ On)
6562, 63, 643syl 18 . . 3 (𝜑 → ran (𝐺𝐴) ⊆ On)
66 onssnum 10077 . . 3 ((ran (𝐺𝐴) ∈ V ∧ ran (𝐺𝐴) ⊆ On) → ran (𝐺𝐴) ∈ dom card)
672, 65, 66sylancr 587 . 2 (𝜑 → ran (𝐺𝐴) ∈ dom card)
68 f1f1orn 6859 . . . 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 6919 . . . 4 (𝑅1𝐴) ∈ V
7170f1oen 9011 . . 3 ((𝐺𝐴):(𝑅1𝐴)–1-1-onto→ran (𝐺𝐴) → (𝑅1𝐴) ≈ ran (𝐺𝐴))
72 ennum 9984 . . 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 257 1 (𝜑 → (𝑅1𝐴) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  wss 3962  ifcif 4530  𝒫 cpw 4604   cuni 4911   class class class wbr 5147  cmpt 5230   E cep 5587  ccnv 5687  dom cdm 5688  ran crn 5689  cima 5691  ccom 5692  Ord word 6384  Oncon0 6385  suc csuc 6387  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  recscrecs 8408   +o coa 8501   ·o comu 8502  cen 8980  OrdIsocoi 9546  harchar 9593  𝑅1cr1 9799  rankcrnk 9800  cardccrd 9972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-oadd 8508  df-omul 8509  df-er 8743  df-en 8984  df-dom 8985  df-oi 9547  df-har 9594  df-r1 9801  df-rank 9802  df-card 9976
This theorem is referenced by:  dfac12r  10184
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