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Theorem rankf 9789
Description: The domain and codomain of the rank function. (Contributed by Mario Carneiro, 28-May-2013.) (Revised by Mario Carneiro, 12-Sep-2013.)
Assertion
Ref Expression
rankf rank:βˆͺ (𝑅1 β€œ On)⟢On

Proof of Theorem rankf
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rank 9760 . . . 4 rank = (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)})
21funmpt2 6588 . . 3 Fun rank
3 mptv 5265 . . . . . 6 (π‘₯ ∈ V ↦ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}) = {⟨π‘₯, π‘§βŸ© ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}}
41, 3eqtri 2761 . . . . 5 rank = {⟨π‘₯, π‘§βŸ© ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}}
54dmeqi 5905 . . . 4 dom rank = dom {⟨π‘₯, π‘§βŸ© ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}}
6 dmopab 5916 . . . . 5 dom {⟨π‘₯, π‘§βŸ© ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}} = {π‘₯ ∣ βˆƒπ‘§ 𝑧 = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}}
7 eqabcb 2876 . . . . . 6 ({π‘₯ ∣ βˆƒπ‘§ 𝑧 = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}} = βˆͺ (𝑅1 β€œ On) ↔ βˆ€π‘₯(βˆƒπ‘§ 𝑧 = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ↔ π‘₯ ∈ βˆͺ (𝑅1 β€œ On)))
8 rankwflemb 9788 . . . . . . 7 (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ↔ βˆƒπ‘¦ ∈ On π‘₯ ∈ (𝑅1β€˜suc 𝑦))
9 intexrab 5341 . . . . . . 7 (βˆƒπ‘¦ ∈ On π‘₯ ∈ (𝑅1β€˜suc 𝑦) ↔ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ∈ V)
10 isset 3488 . . . . . . 7 (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ∈ V ↔ βˆƒπ‘§ 𝑧 = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)})
118, 9, 103bitrri 298 . . . . . 6 (βˆƒπ‘§ 𝑧 = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ↔ π‘₯ ∈ βˆͺ (𝑅1 β€œ On))
127, 11mpgbir 1802 . . . . 5 {π‘₯ ∣ βˆƒπ‘§ 𝑧 = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}} = βˆͺ (𝑅1 β€œ On)
136, 12eqtri 2761 . . . 4 dom {⟨π‘₯, π‘§βŸ© ∣ 𝑧 = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)}} = βˆͺ (𝑅1 β€œ On)
145, 13eqtri 2761 . . 3 dom rank = βˆͺ (𝑅1 β€œ On)
15 df-fn 6547 . . 3 (rank Fn βˆͺ (𝑅1 β€œ On) ↔ (Fun rank ∧ dom rank = βˆͺ (𝑅1 β€œ On)))
162, 14, 15mpbir2an 710 . 2 rank Fn βˆͺ (𝑅1 β€œ On)
17 rabn0 4386 . . . . 5 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} β‰  βˆ… ↔ βˆƒπ‘¦ ∈ On π‘₯ ∈ (𝑅1β€˜suc 𝑦))
188, 17bitr4i 278 . . . 4 (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ↔ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} β‰  βˆ…)
19 intex 5338 . . . . . 6 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} β‰  βˆ… ↔ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ∈ V)
20 vex 3479 . . . . . . 7 π‘₯ ∈ V
211fvmpt2 7010 . . . . . . 7 ((π‘₯ ∈ V ∧ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ∈ V) β†’ (rankβ€˜π‘₯) = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)})
2220, 21mpan 689 . . . . . 6 (∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ∈ V β†’ (rankβ€˜π‘₯) = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)})
2319, 22sylbi 216 . . . . 5 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} β‰  βˆ… β†’ (rankβ€˜π‘₯) = ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)})
24 ssrab2 4078 . . . . . 6 {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} βŠ† On
25 oninton 7783 . . . . . 6 (({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} βŠ† On ∧ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} β‰  βˆ…) β†’ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ∈ On)
2624, 25mpan 689 . . . . 5 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} β‰  βˆ… β†’ ∩ {𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} ∈ On)
2723, 26eqeltrd 2834 . . . 4 ({𝑦 ∈ On ∣ π‘₯ ∈ (𝑅1β€˜suc 𝑦)} β‰  βˆ… β†’ (rankβ€˜π‘₯) ∈ On)
2818, 27sylbi 216 . . 3 (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) β†’ (rankβ€˜π‘₯) ∈ On)
2928rgen 3064 . 2 βˆ€π‘₯ ∈ βˆͺ (𝑅1 β€œ On)(rankβ€˜π‘₯) ∈ On
30 ffnfv 7118 . 2 (rank:βˆͺ (𝑅1 β€œ On)⟢On ↔ (rank Fn βˆͺ (𝑅1 β€œ On) ∧ βˆ€π‘₯ ∈ βˆͺ (𝑅1 β€œ On)(rankβ€˜π‘₯) ∈ On))
3116, 29, 30mpbir2an 710 1 rank:βˆͺ (𝑅1 β€œ On)⟢On
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  {crab 3433  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  βˆͺ cuni 4909  βˆ© cint 4951  {copab 5211   ↦ cmpt 5232  dom cdm 5677   β€œ cima 5680  Oncon0 6365  suc csuc 6367  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  π‘…1cr1 9757  rankcrnk 9758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-r1 9759  df-rank 9760
This theorem is referenced by:  rankon  9790  rankvaln  9794  tcrank  9879  hsmexlem4  10424  hsmexlem5  10425  grur1  10815  aomclem4  41799
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