Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  abfmpunirn Structured version   Visualization version   GIF version

Theorem abfmpunirn 29792
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 3364 . 2 (𝐵 ran 𝐹𝐵 ∈ V)
2 abfmpunirn.2 . . . . . 6 {𝑦𝜑} ∈ V
3 abfmpunirn.1 . . . . . 6 𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})
42, 3fnmpti 6162 . . . . 5 𝐹 Fn 𝑉
5 fnunirn 6654 . . . . 5 (𝐹 Fn 𝑉 → (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ (𝐹𝑥)))
64, 5ax-mp 5 . . . 4 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ (𝐹𝑥))
73fvmpt2 6433 . . . . . . 7 ((𝑥𝑉 ∧ {𝑦𝜑} ∈ V) → (𝐹𝑥) = {𝑦𝜑})
82, 7mpan2 671 . . . . . 6 (𝑥𝑉 → (𝐹𝑥) = {𝑦𝜑})
98eleq2d 2836 . . . . 5 (𝑥𝑉 → (𝐵 ∈ (𝐹𝑥) ↔ 𝐵 ∈ {𝑦𝜑}))
109rexbiia 3188 . . . 4 (∃𝑥𝑉 𝐵 ∈ (𝐹𝑥) ↔ ∃𝑥𝑉 𝐵 ∈ {𝑦𝜑})
116, 10bitri 264 . . 3 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ {𝑦𝜑})
12 abfmpunirn.3 . . . . 5 (𝑦 = 𝐵 → (𝜑𝜓))
1312elabg 3502 . . . 4 (𝐵 ∈ V → (𝐵 ∈ {𝑦𝜑} ↔ 𝜓))
1413rexbidv 3200 . . 3 (𝐵 ∈ V → (∃𝑥𝑉 𝐵 ∈ {𝑦𝜑} ↔ ∃𝑥𝑉 𝜓))
1511, 14syl5bb 272 . 2 (𝐵 ∈ V → (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝜓))
161, 15biadan2 820 1 (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  {cab 2757  wrex 3062  Vcvv 3351   cuni 4574  cmpt 4863  ran crn 5250   Fn wfn 6026  cfv 6031
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
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  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-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  rabfmpunirn  29793  isrnsigaOLD  30515  isrnsiga  30516  isrnmeas  30603
  Copyright terms: Public domain W3C validator