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Theorem rankuni 8670
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 4410 . . . . 5 (𝑥 = 𝐴 𝑥 = 𝐴)
21fveq2d 6152 . . . 4 (𝑥 = 𝐴 → (rank‘ 𝑥) = (rank‘ 𝐴))
3 fveq2 6148 . . . . 5 (𝑥 = 𝐴 → (rank‘𝑥) = (rank‘𝐴))
43unieqd 4412 . . . 4 (𝑥 = 𝐴 (rank‘𝑥) = (rank‘𝐴))
52, 4eqeq12d 2636 . . 3 (𝑥 = 𝐴 → ((rank‘ 𝑥) = (rank‘𝑥) ↔ (rank‘ 𝐴) = (rank‘𝐴)))
6 vex 3189 . . . . . . 7 𝑥 ∈ V
76rankuni2 8662 . . . . . 6 (rank‘ 𝑥) = 𝑧𝑥 (rank‘𝑧)
8 fvex 6158 . . . . . . 7 (rank‘𝑧) ∈ V
98dfiun2 4520 . . . . . 6 𝑧𝑥 (rank‘𝑧) = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)}
107, 9eqtri 2643 . . . . 5 (rank‘ 𝑥) = {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)}
11 df-rex 2913 . . . . . . . 8 (∃𝑧𝑥 𝑦 = (rank‘𝑧) ↔ ∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)))
126rankel 8646 . . . . . . . . . . 11 (𝑧𝑥 → (rank‘𝑧) ∈ (rank‘𝑥))
1312anim1i 591 . . . . . . . . . 10 ((𝑧𝑥𝑦 = (rank‘𝑧)) → ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
1413eximi 1759 . . . . . . . . 9 (∃𝑧(𝑧𝑥𝑦 = (rank‘𝑧)) → ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
15 19.42v 1915 . . . . . . . . . 10 (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ (𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)))
16 eleq1 2686 . . . . . . . . . . . 12 (𝑦 = (rank‘𝑧) → (𝑦 ∈ (rank‘𝑥) ↔ (rank‘𝑧) ∈ (rank‘𝑥)))
1716pm5.32ri 669 . . . . . . . . . . 11 ((𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
1817exbii 1771 . . . . . . . . . 10 (∃𝑧(𝑦 ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)) ↔ ∃𝑧((rank‘𝑧) ∈ (rank‘𝑥) ∧ 𝑦 = (rank‘𝑧)))
19 simpl 473 . . . . . . . . . . 11 ((𝑦 ∈ (rank‘𝑥) ∧ ∃𝑧 𝑦 = (rank‘𝑧)) → 𝑦 ∈ (rank‘𝑥))
20 rankon 8602 . . . . . . . . . . . . . . . . 17 (rank‘𝑥) ∈ On
2120oneli 5794 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ On)
22 r1fnon 8574 . . . . . . . . . . . . . . . . 17 𝑅1 Fn On
23 fndm 5948 . . . . . . . . . . . . . . . . 17 (𝑅1 Fn On → dom 𝑅1 = On)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16 dom 𝑅1 = On
2521, 24syl6eleqr 2709 . . . . . . . . . . . . . . 15 (𝑦 ∈ (rank‘𝑥) → 𝑦 ∈ dom 𝑅1)
26 rankr1id 8669 . . . . . . . . . . . . . . 15 (𝑦 ∈ dom 𝑅1 ↔ (rank‘(𝑅1𝑦)) = 𝑦)
2725, 26sylib 208 . . . . . . . . . . . . . 14 (𝑦 ∈ (rank‘𝑥) → (rank‘(𝑅1𝑦)) = 𝑦)
2827eqcomd 2627 . . . . . . . . . . . . 13 (𝑦 ∈ (rank‘𝑥) → 𝑦 = (rank‘(𝑅1𝑦)))
29 fvex 6158 . . . . . . . . . . . . . 14 (𝑅1𝑦) ∈ V
30 fveq2 6148 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅1𝑦) → (rank‘𝑧) = (rank‘(𝑅1𝑦)))
3130eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑧 = (𝑅1𝑦) → (𝑦 = (rank‘𝑧) ↔ 𝑦 = (rank‘(𝑅1𝑦))))
3229, 31spcev 3286 . . . . . . . . . . . . 13 (𝑦 = (rank‘(𝑅1𝑦)) → ∃𝑧 𝑦 = (rank‘𝑧))
3328, 32syl 17 . . . . . . . . . . . 12 (𝑦 ∈ (rank‘𝑥) → ∃𝑧 𝑦 = (rank‘𝑧))
3433ancli 573 . . . . . . . . . . 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 3656 . . . . . 6 {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
4039unissi 4427 . . . . 5 {𝑦 ∣ ∃𝑧𝑥 𝑦 = (rank‘𝑧)} ⊆ (rank‘𝑥)
4110, 40eqsstri 3614 . . . 4 (rank‘ 𝑥) ⊆ (rank‘𝑥)
42 pwuni 4859 . . . . . . . 8 𝑥 ⊆ 𝒫 𝑥
43 vuniex 6907 . . . . . . . . . 10 𝑥 ∈ V
4443pwex 4808 . . . . . . . . 9 𝒫 𝑥 ∈ V
4544rankss 8656 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝑥 → (rank‘𝑥) ⊆ (rank‘𝒫 𝑥))
4642, 45ax-mp 5 . . . . . . 7 (rank‘𝑥) ⊆ (rank‘𝒫 𝑥)
4743rankpw 8650 . . . . . . 7 (rank‘𝒫 𝑥) = suc (rank‘ 𝑥)
4846, 47sseqtri 3616 . . . . . 6 (rank‘𝑥) ⊆ suc (rank‘ 𝑥)
4948unissi 4427 . . . . 5 (rank‘𝑥) ⊆ suc (rank‘ 𝑥)
50 rankon 8602 . . . . . 6 (rank‘ 𝑥) ∈ On
5150onunisuci 5800 . . . . 5 suc (rank‘ 𝑥) = (rank‘ 𝑥)
5249, 51sseqtri 3616 . . . 4 (rank‘𝑥) ⊆ (rank‘ 𝑥)
5341, 52eqssi 3599 . . 3 (rank‘ 𝑥) = (rank‘𝑥)
545, 53vtoclg 3252 . 2 (𝐴 ∈ V → (rank‘ 𝐴) = (rank‘𝐴))
55 uniexb 6921 . . . . 5 (𝐴 ∈ V ↔ 𝐴 ∈ V)
56 fvprc 6142 . . . . 5 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
5755, 56sylnbi 320 . . . 4 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
58 uni0 4431 . . . 4 ∅ = ∅
5957, 58syl6eqr 2673 . . 3 𝐴 ∈ V → (rank‘ 𝐴) = ∅)
60 fvprc 6142 . . . 4 𝐴 ∈ V → (rank‘𝐴) = ∅)
6160unieqd 4412 . . 3 𝐴 ∈ V → (rank‘𝐴) = ∅)
6259, 61eqtr4d 2658 . 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 384   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wrex 2908  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130   cuni 4402   ciun 4485  dom cdm 5074  Oncon0 5682  suc csuc 5684   Fn wfn 5842  cfv 5847  𝑅1cr1 8569  rankcrnk 8570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-reg 8441  ax-inf2 8482
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-r1 8571  df-rank 8572
This theorem is referenced by:  rankuniss  8673  rankbnd2  8676  rankxplim2  8687  rankxplim3  8688  rankxpsuc  8689  r1limwun  9502  hfuni  31930
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