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

Theorem rabfmpunirn 32670
Description: Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)
Hypotheses
Ref Expression
rabfmpunirn.1 𝐹 = (𝑥𝑉 ↦ {𝑦𝑊𝜑})
rabfmpunirn.2 𝑊 ∈ V
rabfmpunirn.3 (𝑦 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
rabfmpunirn (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 (𝐵𝑊𝜓))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝑦,𝑊   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)   𝑊(𝑥)

Proof of Theorem rabfmpunirn
StepHypRef Expression
1 rabfmpunirn.1 . . . 4 𝐹 = (𝑥𝑉 ↦ {𝑦𝑊𝜑})
2 df-rab 3434 . . . . 5 {𝑦𝑊𝜑} = {𝑦 ∣ (𝑦𝑊𝜑)}
32mpteq2i 5253 . . . 4 (𝑥𝑉 ↦ {𝑦𝑊𝜑}) = (𝑥𝑉 ↦ {𝑦 ∣ (𝑦𝑊𝜑)})
41, 3eqtri 2763 . . 3 𝐹 = (𝑥𝑉 ↦ {𝑦 ∣ (𝑦𝑊𝜑)})
5 rabfmpunirn.2 . . . 4 𝑊 ∈ V
65zfausab 5338 . . 3 {𝑦 ∣ (𝑦𝑊𝜑)} ∈ V
7 eleq1 2827 . . . 4 (𝑦 = 𝐵 → (𝑦𝑊𝐵𝑊))
8 rabfmpunirn.3 . . . 4 (𝑦 = 𝐵 → (𝜑𝜓))
97, 8anbi12d 632 . . 3 (𝑦 = 𝐵 → ((𝑦𝑊𝜑) ↔ (𝐵𝑊𝜓)))
104, 6, 9abfmpunirn 32669 . 2 (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 (𝐵𝑊𝜓)))
11 elex 3499 . . . . 5 (𝐵𝑊𝐵 ∈ V)
1211adantr 480 . . . 4 ((𝐵𝑊𝜓) → 𝐵 ∈ V)
1312rexlimivw 3149 . . 3 (∃𝑥𝑉 (𝐵𝑊𝜓) → 𝐵 ∈ V)
1413pm4.71ri 560 . 2 (∃𝑥𝑉 (𝐵𝑊𝜓) ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 (𝐵𝑊𝜓)))
1510, 14bitr4i 278 1 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 (𝐵𝑊𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  {crab 3433  Vcvv 3478   cuni 4912  cmpt 5231  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator