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Theorem abfmpunirn 29793
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 6163 . . . . 5 𝐹 Fn 𝑉
5 fnunirn 6655 . . . . 5 (𝐹 Fn 𝑉 → (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ (𝐹𝑥)))
64, 5ax-mp 5 . . . 4 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ (𝐹𝑥))
73fvmpt2 6434 . . . . . . 7 ((𝑥𝑉 ∧ {𝑦𝜑} ∈ V) → (𝐹𝑥) = {𝑦𝜑})
82, 7mpan2 665 . . . . . 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 3503 . . . 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 813 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 4575  cmpt 4864  ran crn 5251   Fn wfn 6027  cfv 6032
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 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  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 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-fv 6040
This theorem is referenced by:  rabfmpunirn  29794  isrnsigaOLD  30516  isrnsiga  30517  isrnmeas  30604
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