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Theorem abfmpunirn 32663
Description: Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)
Hypotheses
Ref Expression
abfmpunirn.1 𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})
abfmpunirn.2 {𝑦𝜑} ∈ V
abfmpunirn.3 (𝑦 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
abfmpunirn (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)

Proof of Theorem abfmpunirn
StepHypRef Expression
1 elex 3500 . 2 (𝐵 ran 𝐹𝐵 ∈ V)
2 abfmpunirn.2 . . . . . 6 {𝑦𝜑} ∈ V
3 abfmpunirn.1 . . . . . 6 𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})
42, 3fnmpti 6710 . . . . 5 𝐹 Fn 𝑉
5 fnunirn 7275 . . . . 5 (𝐹 Fn 𝑉 → (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ (𝐹𝑥)))
64, 5ax-mp 5 . . . 4 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ (𝐹𝑥))
73fvmpt2 7026 . . . . . . 7 ((𝑥𝑉 ∧ {𝑦𝜑} ∈ V) → (𝐹𝑥) = {𝑦𝜑})
82, 7mpan2 691 . . . . . 6 (𝑥𝑉 → (𝐹𝑥) = {𝑦𝜑})
98eleq2d 2826 . . . . 5 (𝑥𝑉 → (𝐵 ∈ (𝐹𝑥) ↔ 𝐵 ∈ {𝑦𝜑}))
109rexbiia 3091 . . . 4 (∃𝑥𝑉 𝐵 ∈ (𝐹𝑥) ↔ ∃𝑥𝑉 𝐵 ∈ {𝑦𝜑})
116, 10bitri 275 . . 3 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ {𝑦𝜑})
12 abfmpunirn.3 . . . . 5 (𝑦 = 𝐵 → (𝜑𝜓))
1312elabg 3675 . . . 4 (𝐵 ∈ V → (𝐵 ∈ {𝑦𝜑} ↔ 𝜓))
1413rexbidv 3178 . . 3 (𝐵 ∈ V → (∃𝑥𝑉 𝐵 ∈ {𝑦𝜑} ↔ ∃𝑥𝑉 𝜓))
1511, 14bitrid 283 . 2 (𝐵 ∈ V → (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝜓))
161, 15biadanii 821 1 (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  {cab 2713  wrex 3069  Vcvv 3479   cuni 4906  cmpt 5224  ran crn 5685   Fn wfn 6555  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568
This theorem is referenced by:  rabfmpunirn  32664  isrnsiga  34115  isrnmeas  34202
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