MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac12lem3 Structured version   Visualization version   GIF version

Theorem dfac12lem3 8911
Description: Lemma for dfac12 8915. (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 6158 . . . 4 (𝐺𝐴) ∈ V
21rnex 7047 . . 3 ran (𝐺𝐴) ∈ V
3 ssid 3603 . . . . 5 𝐴𝐴
4 dfac12.1 . . . . . 6 (𝜑𝐴 ∈ On)
5 sseq1 3605 . . . . . . . . 9 (𝑚 = 𝑛 → (𝑚𝐴𝑛𝐴))
6 fveq2 6148 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝐺𝑚) = (𝐺𝑛))
7 f1eq1 6053 . . . . . . . . . . 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 6148 . . . . . . . . . . 11 (𝑚 = 𝑛 → (𝑅1𝑚) = (𝑅1𝑛))
10 f1eq2 6054 . . . . . . . . . . 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 334 . . . . . . . 8 (𝑚 = 𝑛 → ((𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On) ↔ (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)))
1413imbi2d 330 . . . . . . 7 (𝑚 = 𝑛 → ((𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On)) ↔ (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On))))
15 sseq1 3605 . . . . . . . . 9 (𝑚 = 𝐴 → (𝑚𝐴𝐴𝐴))
16 fveq2 6148 . . . . . . . . . . 11 (𝑚 = 𝐴 → (𝐺𝑚) = (𝐺𝐴))
17 f1eq1 6053 . . . . . . . . . . 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 6148 . . . . . . . . . . 11 (𝑚 = 𝐴 → (𝑅1𝑚) = (𝑅1𝐴))
20 f1eq2 6054 . . . . . . . . . . 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 334 . . . . . . . 8 (𝑚 = 𝐴 → ((𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On) ↔ (𝐴𝐴 → (𝐺𝐴):(𝑅1𝐴)–1-1→On)))
2423imbi2d 330 . . . . . . 7 (𝑚 = 𝐴 → ((𝜑 → (𝑚𝐴 → (𝐺𝑚):(𝑅1𝑚)–1-1→On)) ↔ (𝜑 → (𝐴𝐴 → (𝐺𝐴):(𝑅1𝐴)–1-1→On))))
25 r19.21v 2954 . . . . . . . 8 (∀𝑛𝑚 (𝜑 → (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)) ↔ (𝜑 → ∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On)))
26 eloni 5692 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ On → Ord 𝑚)
2726ad2antrl 763 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → Ord 𝑚)
28 ordelss 5698 . . . . . . . . . . . . . . . . 17 ((Ord 𝑚𝑛𝑚) → 𝑛𝑚)
2927, 28sylan 488 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑛𝑚)
30 simplrr 800 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑚𝐴)
3129, 30sstrd 3593 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ 𝑛𝑚) → 𝑛𝐴)
32 pm5.5 351 . . . . . . . . . . . . . . 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 2979 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) → (∀𝑛𝑚 (𝑛𝐴 → (𝐺𝑛):(𝑅1𝑛)–1-1→On) ↔ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On))
354ad2antrr 761 . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . 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 799 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝑚 ∈ On)
40 eqid 2621 . . . . . . . . . . . . . . 15 (OrdIso( E , ran (𝐺 𝑚)) ∘ (𝐺 𝑚)) = (OrdIso( E , ran (𝐺 𝑚)) ∘ (𝐺 𝑚))
41 simplrr 800 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → 𝑚𝐴)
42 simpr 477 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On)
43 fveq2 6148 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝐺𝑛) = (𝐺𝑧))
44 f1eq1 6053 . . . . . . . . . . . . . . . . . . 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 6148 . . . . . . . . . . . . . . . . . . 19 (𝑛 = 𝑧 → (𝑅1𝑛) = (𝑅1𝑧))
47 f1eq2 6054 . . . . . . . . . . . . . . . . . . 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 3159 . . . . . . . . . . . . . . . 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 8910 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑚 ∈ On ∧ 𝑚𝐴)) ∧ ∀𝑛𝑚 (𝐺𝑛):(𝑅1𝑛)–1-1→On) → (𝐺𝑚):(𝑅1𝑚)–1-1→On)
5352ex 450 . . . . . . . . . . . . 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 451 . . . . . . . . 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 7004 . . . . . 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 6058 . . . 4 ((𝐺𝐴):(𝑅1𝐴)–1-1→On → (𝐺𝐴):(𝑅1𝐴)⟶On)
64 frn 6010 . . . 4 ((𝐺𝐴):(𝑅1𝐴)⟶On → ran (𝐺𝐴) ⊆ On)
6562, 63, 643syl 18 . . 3 (𝜑 → ran (𝐺𝐴) ⊆ On)
66 onssnum 8807 . . 3 ((ran (𝐺𝐴) ∈ V ∧ ran (𝐺𝐴) ⊆ On) → ran (𝐺𝐴) ∈ dom card)
672, 65, 66sylancr 694 . 2 (𝜑 → ran (𝐺𝐴) ∈ dom card)
68 f1f1orn 6105 . . . 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 6158 . . . 4 (𝑅1𝐴) ∈ V
7170f1oen 7920 . . 3 ((𝐺𝐴):(𝑅1𝐴)–1-1-onto→ran (𝐺𝐴) → (𝑅1𝐴) ≈ ran (𝐺𝐴))
72 ennum 8717 . . 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 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  wss 3555  ifcif 4058  𝒫 cpw 4130   cuni 4402   class class class wbr 4613  cmpt 4673   E cep 4983  ccnv 5073  dom cdm 5074  ran crn 5075  cima 5077  ccom 5078  Ord word 5681  Oncon0 5682  suc csuc 5684  wf 5843  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  recscrecs 7412   +𝑜 coa 7502   ·𝑜 comu 7503  cen 7896  OrdIsocoi 8358  harchar 8405  𝑅1cr1 8569  rankcrnk 8570  cardccrd 8705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-oadd 7509  df-omul 7510  df-er 7687  df-en 7900  df-dom 7901  df-oi 8359  df-har 8407  df-r1 8571  df-rank 8572  df-card 8709
This theorem is referenced by:  dfac12r  8912
  Copyright terms: Public domain W3C validator