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Theorem abfmpunirn 28832
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 3185 . 2 (𝐵 ran 𝐹𝐵 ∈ V)
2 abfmpunirn.2 . . . . . 6 {𝑦𝜑} ∈ V
3 abfmpunirn.1 . . . . . 6 𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})
42, 3fnmpti 5935 . . . . 5 𝐹 Fn 𝑉
5 fnunirn 6415 . . . . 5 (𝐹 Fn 𝑉 → (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ (𝐹𝑥)))
64, 5ax-mp 5 . . . 4 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ (𝐹𝑥))
73fvmpt2 6200 . . . . . . 7 ((𝑥𝑉 ∧ {𝑦𝜑} ∈ V) → (𝐹𝑥) = {𝑦𝜑})
82, 7mpan2 703 . . . . . 6 (𝑥𝑉 → (𝐹𝑥) = {𝑦𝜑})
98eleq2d 2673 . . . . 5 (𝑥𝑉 → (𝐵 ∈ (𝐹𝑥) ↔ 𝐵 ∈ {𝑦𝜑}))
109rexbiia 3022 . . . 4 (∃𝑥𝑉 𝐵 ∈ (𝐹𝑥) ↔ ∃𝑥𝑉 𝐵 ∈ {𝑦𝜑})
116, 10bitri 263 . . 3 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ {𝑦𝜑})
12 abfmpunirn.3 . . . . 5 (𝑦 = 𝐵 → (𝜑𝜓))
1312elabg 3320 . . . 4 (𝐵 ∈ V → (𝐵 ∈ {𝑦𝜑} ↔ 𝜓))
1413rexbidv 3034 . . 3 (𝐵 ∈ V → (∃𝑥𝑉 𝐵 ∈ {𝑦𝜑} ↔ ∃𝑥𝑉 𝜓))
1511, 14syl5bb 271 . 2 (𝐵 ∈ V → (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝜓))
161, 15biadan2 672 1 (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  {cab 2596  wrex 2897  Vcvv 3173   cuni 4372  cmpt 4643  ran crn 5039   Fn wfn 5799  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812
This theorem is referenced by:  rabfmpunirn  28833  isrnsigaOLD  29502  isrnsiga  29503  isrnmeas  29590
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