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Theorem rabfmpunirn 32409
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 3428 . . . . 5 {𝑦𝑊𝜑} = {𝑦 ∣ (𝑦𝑊𝜑)}
32mpteq2i 5247 . . . 4 (𝑥𝑉 ↦ {𝑦𝑊𝜑}) = (𝑥𝑉 ↦ {𝑦 ∣ (𝑦𝑊𝜑)})
41, 3eqtri 2755 . . 3 𝐹 = (𝑥𝑉 ↦ {𝑦 ∣ (𝑦𝑊𝜑)})
5 rabfmpunirn.2 . . . 4 𝑊 ∈ V
65zfausab 5326 . . 3 {𝑦 ∣ (𝑦𝑊𝜑)} ∈ V
7 eleq1 2816 . . . 4 (𝑦 = 𝐵 → (𝑦𝑊𝐵𝑊))
8 rabfmpunirn.3 . . . 4 (𝑦 = 𝐵 → (𝜑𝜓))
97, 8anbi12d 630 . . 3 (𝑦 = 𝐵 → ((𝑦𝑊𝜑) ↔ (𝐵𝑊𝜓)))
104, 6, 9abfmpunirn 32408 . 2 (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 (𝐵𝑊𝜓)))
11 elex 3488 . . . . 5 (𝐵𝑊𝐵 ∈ V)
1211adantr 480 . . . 4 ((𝐵𝑊𝜓) → 𝐵 ∈ V)
1312rexlimivw 3146 . . 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 205  wa 395   = wceq 1534  wcel 2099  {cab 2704  wrex 3065  {crab 3427  Vcvv 3469   cuni 4903  cmpt 5225  ran crn 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by: (None)
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