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

Theorem rankuni 8609
Description: The rank of a union. Part of Exercise 4 of [Kunen] p. 107. (Contributed by NM, 15-Sep-2006.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankuni (rank‘ 𝐴) = (rank‘𝐴)

Proof of Theorem rankuni
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unieq 4380 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
21fveq2d 6107 . . . 4 (𝑥 = 𝐴 → (rank‘ 𝑥) = (rank‘ 𝐴))
3 fveq2 6103 . . . . 5 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
43unieqd 4382 . . . 4 (𝑥 = 𝐴 (rank‘𝑥) = (rank‘𝐴))
52, 4eqeq12d 2625 . . 3 (𝑥 = 𝐴 → ((rank‘ 𝑥) = (rank‘𝑥) ↔ (rank‘ 𝐴) = (rank‘𝐴)))
6 vex 3176 . . . . . . 7 𝑥 ∈ V
76rankuni2 8601 . . . . . 6 (rank‘ 𝑥) = 𝑧𝑥 (rank‘𝑧)
8 fvex 6113 . . . . . . 7 (rank‘𝑧) ∈ V
98dfiun2 4490 . . . . . 6 𝑧𝑥 (rank‘𝑧) = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)}
107, 9eqtri 2632 . . . . 5 (rank‘ 𝑥) = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)}
11 df-rex 2902 . . . . . . . 8 (∃𝑧𝑥 𝑦 = (rank‘𝑧) ↔ ∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)))
126rankel 8585 . . . . . . . . . . 11 (𝑧𝑥 → (rank‘𝑧) ∈ (rank‘𝑥))
1312anim1i 590 . . . . . . . . . 10 ((𝑧𝑥𝑦 = (rank‘𝑧)) → ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
1413eximi 1752 . . . . . . . . 9 (∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)) → ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
15 19.42v 1905 . . . . . . . . . 10 (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
16 eleq1 2676 . . . . . . . . . . . 12 (𝑦 = (rank‘𝑧) → (𝑦 ∈ (rank‘𝑥) ↔ (rank‘𝑧) ∈ (rank‘𝑥)))
1716pm5.32ri 668 . . . . . . . . . . 11 ((𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
1817exbii 1764 . . . . . . . . . 10 (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
19 simpl 472 . . . . . . . . . . 11 ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
20 rankon 8541 . . . . . . . . . . . . . . . . 17 (rank‘𝑥) ∈ On
2120oneli 5752 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ On)
22 r1fnon 8513 . . . . . . . . . . . . . . . . 17 𝑅1 Fn On
23 fndm 5904 . . . . . . . . . . . . . . . . 17 (𝑅1 Fn On → dom 𝑅1 = On)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16 dom 𝑅1 = On
2521, 24syl6eleqr 2699 . . . . . . . . . . . . . . 15 (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ dom 𝑅1)
26 rankr1id 8608 . . . . . . . . . . . . . . 15 (𝑦 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝑦)) = 𝑦)
2725, 26sylib 207 . . . . . . . . . . . . . 14 (𝑦 ∈ (rank‘𝑥) → (rank‘(𝑅1𝑦)) = 𝑦)
2827eqcomd 2616 . . . . . . . . . . . . 13 (𝑦 ∈ (rank‘𝑥) → 𝑦 = (rank‘(𝑅1𝑦)))
29 fvex 6113 . . . . . . . . . . . . . 14 (𝑅1𝑦) ∈ V
30 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅1𝑦) → (rank‘𝑧) = (rank‘(𝑅1𝑦)))
3130eqeq2d 2620 . . . . . . . . . . . . . 14 (𝑧 = (𝑅1𝑦) → (𝑦 = (rank‘𝑧) ↔ 𝑦 = (rank‘(𝑅1𝑦))))
3229, 31spcev 3273 . . . . . . . . . . . . 13 (𝑦 = (rank‘(𝑅1𝑦)) → ∃𝑧 𝑦 = (rank‘𝑧))
3328, 32syl 17 . . . . . . . . . . . 12 (𝑦 ∈ (rank‘𝑥) → ∃𝑧 𝑦 = (rank‘𝑧))
3433ancli 572 . . . . . . . . . . 11 (𝑦 ∈ (rank‘𝑥) → (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
3519, 34impbii 198 . . . . . . . . . 10 ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
3615, 18, 353bitr3i 289 . . . . . . . . 9 (∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
3714, 36sylib 207 . . . . . . . 8 (∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
3811, 37sylbi 206 . . . . . . 7 (∃𝑧𝑥 𝑦 = (rank‘𝑧) → 𝑦 ∈ (rank‘𝑥))
3938abssi 3640 . . . . . 6 {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
4039unissi 4397 . . . . 5 {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
4110, 40eqsstri 3598 . . . 4 (rank‘ 𝑥) ⊆ (rank‘𝑥)
42 pwuni 4825 . . . . . . . 8 𝑥 ⊆ 𝒫 𝑥
43 vuniex 6852 . . . . . . . . . 10 𝑥 ∈ V
4443pwex 4774 . . . . . . . . 9 𝒫 𝑥 ∈ V
4544rankss 8595 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝑥 → (rank‘𝑥) ⊆ (rank‘𝒫 𝑥))
4642, 45ax-mp 5 . . . . . . 7 (rank‘𝑥) ⊆ (rank‘𝒫 𝑥)
4743rankpw 8589 . . . . . . 7 (rank‘𝒫 𝑥) = suc (rank‘ 𝑥)
4846, 47sseqtri 3600 . . . . . 6 (rank‘𝑥) ⊆ suc (rank‘ 𝑥)
4948unissi 4397 . . . . 5 (rank‘𝑥) ⊆ suc (rank‘ 𝑥)
50 rankon 8541 . . . . . 6 (rank‘ 𝑥) ∈ On
5150onunisuci 5758 . . . . 5 suc (rank‘ 𝑥) = (rank‘ 𝑥)
5249, 51sseqtri 3600 . . . 4 (rank‘𝑥) ⊆ (rank‘ 𝑥)
5341, 52eqssi 3584 . . 3 (rank‘ 𝑥) = (rank‘𝑥)
545, 53vtoclg 3239 . 2 (𝐴 ∈ V → (rank‘ 𝐴) = (rank‘𝐴))
55 uniexb 6866 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ V)
56 fvprc 6097 . . . . 5 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
5755, 56sylnbi 319 . . . 4 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
58 uni0 4401 . . . 4 ∅ = ∅
5957, 58syl6eqr 2662 . . 3 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
60 fvprc 6097 . . . 4 𝐴 ∈ V → (rank‘𝐴) = ∅)
6160unieqd 4382 . . 3 𝐴 ∈ V → (rank‘𝐴) = ∅)
6259, 61eqtr4d 2647 . 2 𝐴 ∈ V → (rank‘ 𝐴) = (rank‘𝐴))
6354, 62pm2.61i 175 1 (rank‘ 𝐴) = (rank‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wrex 2897  Vcvv 3173  wss 3540  c0 3874  𝒫 cpw 4108   cuni 4372   ciun 4455  dom cdm 5038  Oncon0 5640  suc csuc 5642   Fn wfn 5799  cfv 5804  𝑅1cr1 8508  rankcrnk 8509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-r1 8510  df-rank 8511
This theorem is referenced by:  rankuniss  8612  rankbnd2  8615  rankxplim2  8626  rankxplim3  8627  rankxpsuc  8628  r1limwun  9437  hfuni  31461
  Copyright terms: Public domain W3C validator