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Theorem rankuni 8889
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 4580 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
21fveq2d 6336 . . . 4 (𝑥 = 𝐴 → (rank‘ 𝑥) = (rank‘ 𝐴))
3 fveq2 6332 . . . . 5 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
43unieqd 4582 . . . 4 (𝑥 = 𝐴 (rank‘𝑥) = (rank‘𝐴))
52, 4eqeq12d 2785 . . 3 (𝑥 = 𝐴 → ((rank‘ 𝑥) = (rank‘𝑥) ↔ (rank‘ 𝐴) = (rank‘𝐴)))
6 vex 3352 . . . . . . 7 𝑥 ∈ V
76rankuni2 8881 . . . . . 6 (rank‘ 𝑥) = 𝑧𝑥 (rank‘𝑧)
8 fvex 6342 . . . . . . 7 (rank‘𝑧) ∈ V
98dfiun2 4686 . . . . . 6 𝑧𝑥 (rank‘𝑧) = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)}
107, 9eqtri 2792 . . . . 5 (rank‘ 𝑥) = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)}
11 df-rex 3066 . . . . . . . 8 (∃𝑧𝑥 𝑦 = (rank‘𝑧) ↔ ∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)))
126rankel 8865 . . . . . . . . . . 11 (𝑧𝑥 → (rank‘𝑧) ∈ (rank‘𝑥))
1312anim1i 594 . . . . . . . . . 10 ((𝑧𝑥𝑦 = (rank‘𝑧)) → ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
1413eximi 1909 . . . . . . . . 9 (∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)) → ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
15 19.42v 2032 . . . . . . . . . 10 (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
16 eleq1 2837 . . . . . . . . . . . 12 (𝑦 = (rank‘𝑧) → (𝑦 ∈ (rank‘𝑥) ↔ (rank‘𝑧) ∈ (rank‘𝑥)))
1716pm5.32ri 557 . . . . . . . . . . 11 ((𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
1817exbii 1923 . . . . . . . . . 10 (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
19 simpl 468 . . . . . . . . . . 11 ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
20 rankon 8821 . . . . . . . . . . . . . . . . 17 (rank‘𝑥) ∈ On
2120oneli 5978 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ On)
22 r1fnon 8793 . . . . . . . . . . . . . . . . 17 𝑅1 Fn On
23 fndm 6130 . . . . . . . . . . . . . . . . 17 (𝑅1 Fn On → dom 𝑅1 = On)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16 dom 𝑅1 = On
2521, 24syl6eleqr 2860 . . . . . . . . . . . . . . 15 (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ dom 𝑅1)
26 rankr1id 8888 . . . . . . . . . . . . . . 15 (𝑦 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝑦)) = 𝑦)
2725, 26sylib 208 . . . . . . . . . . . . . 14 (𝑦 ∈ (rank‘𝑥) → (rank‘(𝑅1𝑦)) = 𝑦)
2827eqcomd 2776 . . . . . . . . . . . . 13 (𝑦 ∈ (rank‘𝑥) → 𝑦 = (rank‘(𝑅1𝑦)))
29 fvex 6342 . . . . . . . . . . . . . 14 (𝑅1𝑦) ∈ V
30 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅1𝑦) → (rank‘𝑧) = (rank‘(𝑅1𝑦)))
3130eqeq2d 2780 . . . . . . . . . . . . . 14 (𝑧 = (𝑅1𝑦) → (𝑦 = (rank‘𝑧) ↔ 𝑦 = (rank‘(𝑅1𝑦))))
3229, 31spcev 3449 . . . . . . . . . . . . 13 (𝑦 = (rank‘(𝑅1𝑦)) → ∃𝑧 𝑦 = (rank‘𝑧))
3328, 32syl 17 . . . . . . . . . . . 12 (𝑦 ∈ (rank‘𝑥) → ∃𝑧 𝑦 = (rank‘𝑧))
3433ancli 530 . . . . . . . . . . 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 3824 . . . . . 6 {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
4039unissi 4595 . . . . 5 {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
4110, 40eqsstri 3782 . . . 4 (rank‘ 𝑥) ⊆ (rank‘𝑥)
42 pwuni 4608 . . . . . . . 8 𝑥 ⊆ 𝒫 𝑥
43 vuniex 7100 . . . . . . . . . 10 𝑥 ∈ V
4443pwex 4976 . . . . . . . . 9 𝒫 𝑥 ∈ V
4544rankss 8875 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝑥 → (rank‘𝑥) ⊆ (rank‘𝒫 𝑥))
4642, 45ax-mp 5 . . . . . . 7 (rank‘𝑥) ⊆ (rank‘𝒫 𝑥)
4743rankpw 8869 . . . . . . 7 (rank‘𝒫 𝑥) = suc (rank‘ 𝑥)
4846, 47sseqtri 3784 . . . . . 6 (rank‘𝑥) ⊆ suc (rank‘ 𝑥)
4948unissi 4595 . . . . 5 (rank‘𝑥) ⊆ suc (rank‘ 𝑥)
50 rankon 8821 . . . . . 6 (rank‘ 𝑥) ∈ On
5150onunisuci 5984 . . . . 5 suc (rank‘ 𝑥) = (rank‘ 𝑥)
5249, 51sseqtri 3784 . . . 4 (rank‘𝑥) ⊆ (rank‘ 𝑥)
5341, 52eqssi 3766 . . 3 (rank‘ 𝑥) = (rank‘𝑥)
545, 53vtoclg 3415 . 2 (𝐴 ∈ V → (rank‘ 𝐴) = (rank‘𝐴))
55 uniexb 7119 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ V)
56 fvprc 6326 . . . . 5 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
5755, 56sylnbi 319 . . . 4 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
58 uni0 4599 . . . 4 ∅ = ∅
5957, 58syl6eqr 2822 . . 3 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
60 fvprc 6326 . . . 4 𝐴 ∈ V → (rank‘𝐴) = ∅)
6160unieqd 4582 . . 3 𝐴 ∈ V → (rank‘𝐴) = ∅)
6259, 61eqtr4d 2807 . 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 382   = wceq 1630  wex 1851  wcel 2144  {cab 2756  wrex 3061  Vcvv 3349  wss 3721  c0 4061  𝒫 cpw 4295   cuni 4572   ciun 4652  dom cdm 5249  Oncon0 5866  suc csuc 5868   Fn wfn 6026  cfv 6031  𝑅1cr1 8788  rankcrnk 8789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-reg 8652  ax-inf2 8701
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-r1 8790  df-rank 8791
This theorem is referenced by:  rankuniss  8892  rankbnd2  8895  rankxplim2  8906  rankxplim3  8907  rankxpsuc  8908  r1limwun  9759  hfuni  32622
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