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Theorem rankuni 8891
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 4588 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
21fveq2d 6348 . . . 4 (𝑥 = 𝐴 → (rank‘ 𝑥) = (rank‘ 𝐴))
3 fveq2 6344 . . . . 5 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
43unieqd 4590 . . . 4 (𝑥 = 𝐴 (rank‘𝑥) = (rank‘𝐴))
52, 4eqeq12d 2767 . . 3 (𝑥 = 𝐴 → ((rank‘ 𝑥) = (rank‘𝑥) ↔ (rank‘ 𝐴) = (rank‘𝐴)))
6 vex 3335 . . . . . . 7 𝑥 ∈ V
76rankuni2 8883 . . . . . 6 (rank‘ 𝑥) = 𝑧𝑥 (rank‘𝑧)
8 fvex 6354 . . . . . . 7 (rank‘𝑧) ∈ V
98dfiun2 4698 . . . . . 6 𝑧𝑥 (rank‘𝑧) = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)}
107, 9eqtri 2774 . . . . 5 (rank‘ 𝑥) = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)}
11 df-rex 3048 . . . . . . . 8 (∃𝑧𝑥 𝑦 = (rank‘𝑧) ↔ ∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)))
126rankel 8867 . . . . . . . . . . 11 (𝑧𝑥 → (rank‘𝑧) ∈ (rank‘𝑥))
1312anim1i 593 . . . . . . . . . 10 ((𝑧𝑥𝑦 = (rank‘𝑧)) → ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
1413eximi 1903 . . . . . . . . 9 (∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)) → ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
15 19.42v 2022 . . . . . . . . . 10 (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
16 eleq1 2819 . . . . . . . . . . . 12 (𝑦 = (rank‘𝑧) → (𝑦 ∈ (rank‘𝑥) ↔ (rank‘𝑧) ∈ (rank‘𝑥)))
1716pm5.32ri 673 . . . . . . . . . . 11 ((𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
1817exbii 1915 . . . . . . . . . 10 (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
19 simpl 474 . . . . . . . . . . 11 ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
20 rankon 8823 . . . . . . . . . . . . . . . . 17 (rank‘𝑥) ∈ On
2120oneli 5988 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ On)
22 r1fnon 8795 . . . . . . . . . . . . . . . . 17 𝑅1 Fn On
23 fndm 6143 . . . . . . . . . . . . . . . . 17 (𝑅1 Fn On → dom 𝑅1 = On)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16 dom 𝑅1 = On
2521, 24syl6eleqr 2842 . . . . . . . . . . . . . . 15 (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ dom 𝑅1)
26 rankr1id 8890 . . . . . . . . . . . . . . 15 (𝑦 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝑦)) = 𝑦)
2725, 26sylib 208 . . . . . . . . . . . . . 14 (𝑦 ∈ (rank‘𝑥) → (rank‘(𝑅1𝑦)) = 𝑦)
2827eqcomd 2758 . . . . . . . . . . . . 13 (𝑦 ∈ (rank‘𝑥) → 𝑦 = (rank‘(𝑅1𝑦)))
29 fvex 6354 . . . . . . . . . . . . . 14 (𝑅1𝑦) ∈ V
30 fveq2 6344 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅1𝑦) → (rank‘𝑧) = (rank‘(𝑅1𝑦)))
3130eqeq2d 2762 . . . . . . . . . . . . . 14 (𝑧 = (𝑅1𝑦) → (𝑦 = (rank‘𝑧) ↔ 𝑦 = (rank‘(𝑅1𝑦))))
3229, 31spcev 3432 . . . . . . . . . . . . 13 (𝑦 = (rank‘(𝑅1𝑦)) → ∃𝑧 𝑦 = (rank‘𝑧))
3328, 32syl 17 . . . . . . . . . . . 12 (𝑦 ∈ (rank‘𝑥) → ∃𝑧 𝑦 = (rank‘𝑧))
3433ancli 575 . . . . . . . . . . 11 (𝑦 ∈ (rank‘𝑥) → (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
3519, 34impbii 199 . . . . . . . . . 10 ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
3615, 18, 353bitr3i 290 . . . . . . . . 9 (∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ 𝑦 ∈ (rank‘𝑥))
3714, 36sylib 208 . . . . . . . 8 (∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
3811, 37sylbi 207 . . . . . . 7 (∃𝑧𝑥 𝑦 = (rank‘𝑧) → 𝑦 ∈ (rank‘𝑥))
3938abssi 3810 . . . . . 6 {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
4039unissi 4605 . . . . 5 {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
4110, 40eqsstri 3768 . . . 4 (rank‘ 𝑥) ⊆ (rank‘𝑥)
42 pwuni 4618 . . . . . . . 8 𝑥 ⊆ 𝒫 𝑥
43 vuniex 7111 . . . . . . . . . 10 𝑥 ∈ V
4443pwex 4989 . . . . . . . . 9 𝒫 𝑥 ∈ V
4544rankss 8877 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝑥 → (rank‘𝑥) ⊆ (rank‘𝒫 𝑥))
4642, 45ax-mp 5 . . . . . . 7 (rank‘𝑥) ⊆ (rank‘𝒫 𝑥)
4743rankpw 8871 . . . . . . 7 (rank‘𝒫 𝑥) = suc (rank‘ 𝑥)
4846, 47sseqtri 3770 . . . . . 6 (rank‘𝑥) ⊆ suc (rank‘ 𝑥)
4948unissi 4605 . . . . 5 (rank‘𝑥) ⊆ suc (rank‘ 𝑥)
50 rankon 8823 . . . . . 6 (rank‘ 𝑥) ∈ On
5150onunisuci 5994 . . . . 5 suc (rank‘ 𝑥) = (rank‘ 𝑥)
5249, 51sseqtri 3770 . . . 4 (rank‘𝑥) ⊆ (rank‘ 𝑥)
5341, 52eqssi 3752 . . 3 (rank‘ 𝑥) = (rank‘𝑥)
545, 53vtoclg 3398 . 2 (𝐴 ∈ V → (rank‘ 𝐴) = (rank‘𝐴))
55 uniexb 7130 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ V)
56 fvprc 6338 . . . . 5 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
5755, 56sylnbi 319 . . . 4 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
58 uni0 4609 . . . 4 ∅ = ∅
5957, 58syl6eqr 2804 . . 3 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
60 fvprc 6338 . . . 4 𝐴 ∈ V → (rank‘𝐴) = ∅)
6160unieqd 4590 . . 3 𝐴 ∈ V → (rank‘𝐴) = ∅)
6259, 61eqtr4d 2789 . 2 𝐴 ∈ V → (rank‘ 𝐴) = (rank‘𝐴))
6354, 62pm2.61i 176 1 (rank‘ 𝐴) = (rank‘𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1624  wex 1845  wcel 2131  {cab 2738  wrex 3043  Vcvv 3332  wss 3707  c0 4050  𝒫 cpw 4294   cuni 4580   ciun 4664  dom cdm 5258  Oncon0 5876  suc csuc 5878   Fn wfn 6036  cfv 6041  𝑅1cr1 8790  rankcrnk 8791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-reg 8654  ax-inf2 8703
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-pss 3723  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-tp 4318  df-op 4320  df-uni 4581  df-int 4620  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-tr 4897  df-id 5166  df-eprel 5171  df-po 5179  df-so 5180  df-fr 5217  df-we 5219  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-pred 5833  df-ord 5879  df-on 5880  df-lim 5881  df-suc 5882  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-om 7223  df-wrecs 7568  df-recs 7629  df-rdg 7667  df-r1 8792  df-rank 8793
This theorem is referenced by:  rankuniss  8894  rankbnd2  8897  rankxplim2  8908  rankxplim3  8909  rankxpsuc  8910  r1limwun  9742  hfuni  32589
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