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Theorem rankuni 9858
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 (rankβ€˜βˆͺ 𝐴) = βˆͺ (rankβ€˜π΄)

Proof of Theorem rankuni
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unieq 4920 . . . . 5 (π‘₯ = 𝐴 β†’ βˆͺ π‘₯ = βˆͺ 𝐴)
21fveq2d 6896 . . . 4 (π‘₯ = 𝐴 β†’ (rankβ€˜βˆͺ π‘₯) = (rankβ€˜βˆͺ 𝐴))
3 fveq2 6892 . . . . 5 (π‘₯ = 𝐴 β†’ (rankβ€˜π‘₯) = (rankβ€˜π΄))
43unieqd 4923 . . . 4 (π‘₯ = 𝐴 β†’ βˆͺ (rankβ€˜π‘₯) = βˆͺ (rankβ€˜π΄))
52, 4eqeq12d 2749 . . 3 (π‘₯ = 𝐴 β†’ ((rankβ€˜βˆͺ π‘₯) = βˆͺ (rankβ€˜π‘₯) ↔ (rankβ€˜βˆͺ 𝐴) = βˆͺ (rankβ€˜π΄)))
6 vex 3479 . . . . . . 7 π‘₯ ∈ V
76rankuni2 9850 . . . . . 6 (rankβ€˜βˆͺ π‘₯) = βˆͺ 𝑧 ∈ π‘₯ (rankβ€˜π‘§)
8 fvex 6905 . . . . . . 7 (rankβ€˜π‘§) ∈ V
98dfiun2 5037 . . . . . 6 βˆͺ 𝑧 ∈ π‘₯ (rankβ€˜π‘§) = βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑦 = (rankβ€˜π‘§)}
107, 9eqtri 2761 . . . . 5 (rankβ€˜βˆͺ π‘₯) = βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑦 = (rankβ€˜π‘§)}
11 df-rex 3072 . . . . . . . 8 (βˆƒπ‘§ ∈ π‘₯ 𝑦 = (rankβ€˜π‘§) ↔ βˆƒπ‘§(𝑧 ∈ π‘₯ ∧ 𝑦 = (rankβ€˜π‘§)))
126rankel 9834 . . . . . . . . . . 11 (𝑧 ∈ π‘₯ β†’ (rankβ€˜π‘§) ∈ (rankβ€˜π‘₯))
1312anim1i 616 . . . . . . . . . 10 ((𝑧 ∈ π‘₯ ∧ 𝑦 = (rankβ€˜π‘§)) β†’ ((rankβ€˜π‘§) ∈ (rankβ€˜π‘₯) ∧ 𝑦 = (rankβ€˜π‘§)))
1413eximi 1838 . . . . . . . . 9 (βˆƒπ‘§(𝑧 ∈ π‘₯ ∧ 𝑦 = (rankβ€˜π‘§)) β†’ βˆƒπ‘§((rankβ€˜π‘§) ∈ (rankβ€˜π‘₯) ∧ 𝑦 = (rankβ€˜π‘§)))
15 19.42v 1958 . . . . . . . . . 10 (βˆƒπ‘§(𝑦 ∈ (rankβ€˜π‘₯) ∧ 𝑦 = (rankβ€˜π‘§)) ↔ (𝑦 ∈ (rankβ€˜π‘₯) ∧ βˆƒπ‘§ 𝑦 = (rankβ€˜π‘§)))
16 eleq1 2822 . . . . . . . . . . . 12 (𝑦 = (rankβ€˜π‘§) β†’ (𝑦 ∈ (rankβ€˜π‘₯) ↔ (rankβ€˜π‘§) ∈ (rankβ€˜π‘₯)))
1716pm5.32ri 577 . . . . . . . . . . 11 ((𝑦 ∈ (rankβ€˜π‘₯) ∧ 𝑦 = (rankβ€˜π‘§)) ↔ ((rankβ€˜π‘§) ∈ (rankβ€˜π‘₯) ∧ 𝑦 = (rankβ€˜π‘§)))
1817exbii 1851 . . . . . . . . . 10 (βˆƒπ‘§(𝑦 ∈ (rankβ€˜π‘₯) ∧ 𝑦 = (rankβ€˜π‘§)) ↔ βˆƒπ‘§((rankβ€˜π‘§) ∈ (rankβ€˜π‘₯) ∧ 𝑦 = (rankβ€˜π‘§)))
19 simpl 484 . . . . . . . . . . 11 ((𝑦 ∈ (rankβ€˜π‘₯) ∧ βˆƒπ‘§ 𝑦 = (rankβ€˜π‘§)) β†’ 𝑦 ∈ (rankβ€˜π‘₯))
20 rankon 9790 . . . . . . . . . . . . . . . . 17 (rankβ€˜π‘₯) ∈ On
2120oneli 6479 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (rankβ€˜π‘₯) β†’ 𝑦 ∈ On)
22 r1fnon 9762 . . . . . . . . . . . . . . . . 17 𝑅1 Fn On
23 fndm 6653 . . . . . . . . . . . . . . . . 17 (𝑅1 Fn On β†’ dom 𝑅1 = On)
2422, 23ax-mp 5 . . . . . . . . . . . . . . . 16 dom 𝑅1 = On
2521, 24eleqtrrdi 2845 . . . . . . . . . . . . . . 15 (𝑦 ∈ (rankβ€˜π‘₯) β†’ 𝑦 ∈ dom 𝑅1)
26 rankr1id 9857 . . . . . . . . . . . . . . 15 (𝑦 ∈ dom 𝑅1 ↔ (rankβ€˜(𝑅1β€˜π‘¦)) = 𝑦)
2725, 26sylib 217 . . . . . . . . . . . . . 14 (𝑦 ∈ (rankβ€˜π‘₯) β†’ (rankβ€˜(𝑅1β€˜π‘¦)) = 𝑦)
2827eqcomd 2739 . . . . . . . . . . . . 13 (𝑦 ∈ (rankβ€˜π‘₯) β†’ 𝑦 = (rankβ€˜(𝑅1β€˜π‘¦)))
29 fvex 6905 . . . . . . . . . . . . . 14 (𝑅1β€˜π‘¦) ∈ V
30 fveq2 6892 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅1β€˜π‘¦) β†’ (rankβ€˜π‘§) = (rankβ€˜(𝑅1β€˜π‘¦)))
3130eqeq2d 2744 . . . . . . . . . . . . . 14 (𝑧 = (𝑅1β€˜π‘¦) β†’ (𝑦 = (rankβ€˜π‘§) ↔ 𝑦 = (rankβ€˜(𝑅1β€˜π‘¦))))
3229, 31spcev 3597 . . . . . . . . . . . . 13 (𝑦 = (rankβ€˜(𝑅1β€˜π‘¦)) β†’ βˆƒπ‘§ 𝑦 = (rankβ€˜π‘§))
3328, 32syl 17 . . . . . . . . . . . 12 (𝑦 ∈ (rankβ€˜π‘₯) β†’ βˆƒπ‘§ 𝑦 = (rankβ€˜π‘§))
3433ancli 550 . . . . . . . . . . 11 (𝑦 ∈ (rankβ€˜π‘₯) β†’ (𝑦 ∈ (rankβ€˜π‘₯) ∧ βˆƒπ‘§ 𝑦 = (rankβ€˜π‘§)))
3519, 34impbii 208 . . . . . . . . . 10 ((𝑦 ∈ (rankβ€˜π‘₯) ∧ βˆƒπ‘§ 𝑦 = (rankβ€˜π‘§)) ↔ 𝑦 ∈ (rankβ€˜π‘₯))
3615, 18, 353bitr3i 301 . . . . . . . . 9 (βˆƒπ‘§((rankβ€˜π‘§) ∈ (rankβ€˜π‘₯) ∧ 𝑦 = (rankβ€˜π‘§)) ↔ 𝑦 ∈ (rankβ€˜π‘₯))
3714, 36sylib 217 . . . . . . . 8 (βˆƒπ‘§(𝑧 ∈ π‘₯ ∧ 𝑦 = (rankβ€˜π‘§)) β†’ 𝑦 ∈ (rankβ€˜π‘₯))
3811, 37sylbi 216 . . . . . . 7 (βˆƒπ‘§ ∈ π‘₯ 𝑦 = (rankβ€˜π‘§) β†’ 𝑦 ∈ (rankβ€˜π‘₯))
3938abssi 4068 . . . . . 6 {𝑦 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑦 = (rankβ€˜π‘§)} βŠ† (rankβ€˜π‘₯)
4039unissi 4918 . . . . 5 βˆͺ {𝑦 ∣ βˆƒπ‘§ ∈ π‘₯ 𝑦 = (rankβ€˜π‘§)} βŠ† βˆͺ (rankβ€˜π‘₯)
4110, 40eqsstri 4017 . . . 4 (rankβ€˜βˆͺ π‘₯) βŠ† βˆͺ (rankβ€˜π‘₯)
42 pwuni 4950 . . . . . . . 8 π‘₯ βŠ† 𝒫 βˆͺ π‘₯
43 vuniex 7729 . . . . . . . . . 10 βˆͺ π‘₯ ∈ V
4443pwex 5379 . . . . . . . . 9 𝒫 βˆͺ π‘₯ ∈ V
4544rankss 9844 . . . . . . . 8 (π‘₯ βŠ† 𝒫 βˆͺ π‘₯ β†’ (rankβ€˜π‘₯) βŠ† (rankβ€˜π’« βˆͺ π‘₯))
4642, 45ax-mp 5 . . . . . . 7 (rankβ€˜π‘₯) βŠ† (rankβ€˜π’« βˆͺ π‘₯)
4743rankpw 9838 . . . . . . 7 (rankβ€˜π’« βˆͺ π‘₯) = suc (rankβ€˜βˆͺ π‘₯)
4846, 47sseqtri 4019 . . . . . 6 (rankβ€˜π‘₯) βŠ† suc (rankβ€˜βˆͺ π‘₯)
4948unissi 4918 . . . . 5 βˆͺ (rankβ€˜π‘₯) βŠ† βˆͺ suc (rankβ€˜βˆͺ π‘₯)
50 rankon 9790 . . . . . 6 (rankβ€˜βˆͺ π‘₯) ∈ On
5150onunisuci 6485 . . . . 5 βˆͺ suc (rankβ€˜βˆͺ π‘₯) = (rankβ€˜βˆͺ π‘₯)
5249, 51sseqtri 4019 . . . 4 βˆͺ (rankβ€˜π‘₯) βŠ† (rankβ€˜βˆͺ π‘₯)
5341, 52eqssi 3999 . . 3 (rankβ€˜βˆͺ π‘₯) = βˆͺ (rankβ€˜π‘₯)
545, 53vtoclg 3557 . 2 (𝐴 ∈ V β†’ (rankβ€˜βˆͺ 𝐴) = βˆͺ (rankβ€˜π΄))
55 uniexb 7751 . . . . 5 (𝐴 ∈ V ↔ βˆͺ 𝐴 ∈ V)
56 fvprc 6884 . . . . 5 (Β¬ βˆͺ 𝐴 ∈ V β†’ (rankβ€˜βˆͺ 𝐴) = βˆ…)
5755, 56sylnbi 330 . . . 4 (Β¬ 𝐴 ∈ V β†’ (rankβ€˜βˆͺ 𝐴) = βˆ…)
58 uni0 4940 . . . 4 βˆͺ βˆ… = βˆ…
5957, 58eqtr4di 2791 . . 3 (Β¬ 𝐴 ∈ V β†’ (rankβ€˜βˆͺ 𝐴) = βˆͺ βˆ…)
60 fvprc 6884 . . . 4 (Β¬ 𝐴 ∈ V β†’ (rankβ€˜π΄) = βˆ…)
6160unieqd 4923 . . 3 (Β¬ 𝐴 ∈ V β†’ βˆͺ (rankβ€˜π΄) = βˆͺ βˆ…)
6259, 61eqtr4d 2776 . 2 (Β¬ 𝐴 ∈ V β†’ (rankβ€˜βˆͺ 𝐴) = βˆͺ (rankβ€˜π΄))
6354, 62pm2.61i 182 1 (rankβ€˜βˆͺ 𝐴) = βˆͺ (rankβ€˜π΄)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆƒwrex 3071  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ cuni 4909  βˆͺ ciun 4998  dom cdm 5677  Oncon0 6365  suc csuc 6367   Fn wfn 6539  β€˜cfv 6544  π‘…1cr1 9757  rankcrnk 9758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-reg 9587  ax-inf2 9636
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-r1 9759  df-rank 9760
This theorem is referenced by:  rankuniss  9861  rankbnd2  9864  rankxplim2  9875  rankxplim3  9876  rankxpsuc  9877  r1limwun  10731  hfuni  35156
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