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Theorem rankuni 9906
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 4924 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
21fveq2d 6905 . . . 4 (𝑥 = 𝐴 → (rank‘ 𝑥) = (rank‘ 𝐴))
3 fveq2 6901 . . . . 5 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
43unieqd 4926 . . . 4 (𝑥 = 𝐴 (rank‘𝑥) = (rank‘𝐴))
52, 4eqeq12d 2742 . . 3 (𝑥 = 𝐴 → ((rank‘ 𝑥) = (rank‘𝑥) ↔ (rank‘ 𝐴) = (rank‘𝐴)))
6 vex 3466 . . . . . . 7 𝑥 ∈ V
76rankuni2 9898 . . . . . 6 (rank‘ 𝑥) = 𝑧𝑥 (rank‘𝑧)
8 fvex 6914 . . . . . . 7 (rank‘𝑧) ∈ V
98dfiun2 5041 . . . . . 6 𝑧𝑥 (rank‘𝑧) = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)}
107, 9eqtri 2754 . . . . 5 (rank‘ 𝑥) = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)}
11 df-rex 3061 . . . . . . . 8 (∃𝑧𝑥 𝑦 = (rank‘𝑧) ↔ ∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)))
126rankel 9882 . . . . . . . . . . 11 (𝑧𝑥 → (rank‘𝑧) ∈ (rank‘𝑥))
1312anim1i 613 . . . . . . . . . 10 ((𝑧𝑥𝑦 = (rank‘𝑧)) → ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
1413eximi 1830 . . . . . . . . 9 (∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)) → ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
15 19.42v 1950 . . . . . . . . . 10 (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
16 eleq1 2814 . . . . . . . . . . . 12 (𝑦 = (rank‘𝑧) → (𝑦 ∈ (rank‘𝑥) ↔ (rank‘𝑧) ∈ (rank‘𝑥)))
1716pm5.32ri 574 . . . . . . . . . . 11 ((𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
1817exbii 1843 . . . . . . . . . 10 (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
19 simpl 481 . . . . . . . . . . 11 ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
20 rankon 9838 . . . . . . . . . . . . . . . . 17 (rank‘𝑥) ∈ On
2120oneli 6490 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ On)
22 r1fnon 9810 . . . . . . . . . . . . . . . . 17 𝑅1 Fn On
23 fndm 6663 . . . . . . . . . . . . . . . . 17 (𝑅1 Fn On → dom 𝑅1 = On)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16 dom 𝑅1 = On
2521, 24eleqtrrdi 2837 . . . . . . . . . . . . . . 15 (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ dom 𝑅1)
26 rankr1id 9905 . . . . . . . . . . . . . . 15 (𝑦 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝑦)) = 𝑦)
2725, 26sylib 217 . . . . . . . . . . . . . 14 (𝑦 ∈ (rank‘𝑥) → (rank‘(𝑅1𝑦)) = 𝑦)
2827eqcomd 2732 . . . . . . . . . . . . 13 (𝑦 ∈ (rank‘𝑥) → 𝑦 = (rank‘(𝑅1𝑦)))
29 fvex 6914 . . . . . . . . . . . . . 14 (𝑅1𝑦) ∈ V
30 fveq2 6901 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅1𝑦) → (rank‘𝑧) = (rank‘(𝑅1𝑦)))
3130eqeq2d 2737 . . . . . . . . . . . . . 14 (𝑧 = (𝑅1𝑦) → (𝑦 = (rank‘𝑧) ↔ 𝑦 = (rank‘(𝑅1𝑦))))
3229, 31spcev 3592 . . . . . . . . . . . . 13 (𝑦 = (rank‘(𝑅1𝑦)) → ∃𝑧 𝑦 = (rank‘𝑧))
3328, 32syl 17 . . . . . . . . . . . 12 (𝑦 ∈ (rank‘𝑥) → ∃𝑧 𝑦 = (rank‘𝑧))
3433ancli 547 . . . . . . . . . . 11 (𝑦 ∈ (rank‘𝑥) → (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
3519, 34impbii 208 . . . . . . . . . 10 ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
3615, 18, 353bitr3i 300 . . . . . . . . 9 (∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
3714, 36sylib 217 . . . . . . . 8 (∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
3811, 37sylbi 216 . . . . . . 7 (∃𝑧𝑥 𝑦 = (rank‘𝑧) → 𝑦 ∈ (rank‘𝑥))
3938abssi 4066 . . . . . 6 {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
4039unissi 4922 . . . . 5 {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
4110, 40eqsstri 4014 . . . 4 (rank‘ 𝑥) ⊆ (rank‘𝑥)
42 pwuni 4953 . . . . . . . 8 𝑥 ⊆ 𝒫 𝑥
43 vuniex 7750 . . . . . . . . . 10 𝑥 ∈ V
4443pwex 5384 . . . . . . . . 9 𝒫 𝑥 ∈ V
4544rankss 9892 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝑥 → (rank‘𝑥) ⊆ (rank‘𝒫 𝑥))
4642, 45ax-mp 5 . . . . . . 7 (rank‘𝑥) ⊆ (rank‘𝒫 𝑥)
4743rankpw 9886 . . . . . . 7 (rank‘𝒫 𝑥) = suc (rank‘ 𝑥)
4846, 47sseqtri 4016 . . . . . 6 (rank‘𝑥) ⊆ suc (rank‘ 𝑥)
4948unissi 4922 . . . . 5 (rank‘𝑥) ⊆ suc (rank‘ 𝑥)
50 rankon 9838 . . . . . 6 (rank‘ 𝑥) ∈ On
5150onunisuci 6496 . . . . 5 suc (rank‘ 𝑥) = (rank‘ 𝑥)
5249, 51sseqtri 4016 . . . 4 (rank‘𝑥) ⊆ (rank‘ 𝑥)
5341, 52eqssi 3996 . . 3 (rank‘ 𝑥) = (rank‘𝑥)
545, 53vtoclg 3534 . 2 (𝐴 ∈ V → (rank‘ 𝐴) = (rank‘𝐴))
55 uniexb 7772 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ V)
56 fvprc 6893 . . . . 5 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
5755, 56sylnbi 329 . . . 4 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
58 uni0 4943 . . . 4 ∅ = ∅
5957, 58eqtr4di 2784 . . 3 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
60 fvprc 6893 . . . 4 𝐴 ∈ V → (rank‘𝐴) = ∅)
6160unieqd 4926 . . 3 𝐴 ∈ V → (rank‘𝐴) = ∅)
6259, 61eqtr4d 2769 . 2 𝐴 ∈ V → (rank‘ 𝐴) = (rank‘𝐴))
6354, 62pm2.61i 182 1 (rank‘ 𝐴) = (rank‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 394   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wrex 3060  Vcvv 3462  wss 3947  c0 4325  𝒫 cpw 4607   cuni 4913   ciun 5001  dom cdm 5682  Oncon0 6376  suc csuc 6378   Fn wfn 6549  cfv 6554  𝑅1cr1 9805  rankcrnk 9806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-reg 9635  ax-inf2 9684
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-int 4955  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-r1 9807  df-rank 9808
This theorem is referenced by:  rankuniss  9909  rankbnd2  9912  rankxplim2  9923  rankxplim3  9924  rankxpsuc  9925  r1limwun  10779  hfuni  36008
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