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Theorem fobigcup 32344
Description: Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
fobigcup Bigcup :V–onto→V

Proof of Theorem fobigcup
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniexg 7102 . . . 4 (𝑥 ∈ V → 𝑥 ∈ V)
21rgen 3071 . . 3 𝑥 ∈ V 𝑥 ∈ V
3 dfbigcup2 32343 . . . 4 Bigcup = (𝑥 ∈ V ↦ 𝑥)
43mptfng 6159 . . 3 (∀𝑥 ∈ V 𝑥 ∈ V ↔ Bigcup Fn V)
52, 4mpbi 220 . 2 Bigcup Fn V
63rnmpt 5509 . . 3 ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝑥}
7 vex 3354 . . . . 5 𝑦 ∈ V
8 snex 5036 . . . . . 6 {𝑦} ∈ V
97unisn 4589 . . . . . . 7 {𝑦} = 𝑦
109eqcomi 2780 . . . . . 6 𝑦 = {𝑦}
11 unieq 4582 . . . . . . . 8 (𝑥 = {𝑦} → 𝑥 = {𝑦})
1211eqeq2d 2781 . . . . . . 7 (𝑥 = {𝑦} → (𝑦 = 𝑥𝑦 = {𝑦}))
1312rspcev 3460 . . . . . 6 (({𝑦} ∈ V ∧ 𝑦 = {𝑦}) → ∃𝑥 ∈ V 𝑦 = 𝑥)
148, 10, 13mp2an 672 . . . . 5 𝑥 ∈ V 𝑦 = 𝑥
157, 142th 254 . . . 4 (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = 𝑥)
1615abbi2i 2887 . . 3 V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = 𝑥}
176, 16eqtr4i 2796 . 2 ran Bigcup = V
18 df-fo 6037 . 2 ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V))
195, 17, 18mpbir2an 690 1 Bigcup :V–onto→V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  {cab 2757  wral 3061  wrex 3062  Vcvv 3351  {csn 4316   cuni 4574  ran crn 5250   Fn wfn 6026  ontowfo 6029   Bigcup cbigcup 32278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-symdif 3993  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-eprel 5162  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-1st 7315  df-2nd 7316  df-txp 32298  df-bigcup 32302
This theorem is referenced by:  fnbigcup  32345
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