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Theorem rabfmpunirn 30406
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 3139 . . . . 5 {𝑦𝑊𝜑} = {𝑦 ∣ (𝑦𝑊𝜑)}
32mpteq2i 5134 . . . 4 (𝑥𝑉 ↦ {𝑦𝑊𝜑}) = (𝑥𝑉 ↦ {𝑦 ∣ (𝑦𝑊𝜑)})
41, 3eqtri 2845 . . 3 𝐹 = (𝑥𝑉 ↦ {𝑦 ∣ (𝑦𝑊𝜑)})
5 rabfmpunirn.2 . . . 4 𝑊 ∈ V
65zfausab 5209 . . 3 {𝑦 ∣ (𝑦𝑊𝜑)} ∈ V
7 eleq1 2901 . . . 4 (𝑦 = 𝐵 → (𝑦𝑊𝐵𝑊))
8 rabfmpunirn.3 . . . 4 (𝑦 = 𝐵 → (𝜑𝜓))
97, 8anbi12d 633 . . 3 (𝑦 = 𝐵 → ((𝑦𝑊𝜑) ↔ (𝐵𝑊𝜓)))
104, 6, 9abfmpunirn 30405 . 2 (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 (𝐵𝑊𝜓)))
11 elex 3487 . . . . 5 (𝐵𝑊𝐵 ∈ V)
1211adantr 484 . . . 4 ((𝐵𝑊𝜓) → 𝐵 ∈ V)
1312rexlimivw 3268 . . 3 (∃𝑥𝑉 (𝐵𝑊𝜓) → 𝐵 ∈ V)
1413pm4.71ri 564 . 2 (∃𝑥𝑉 (𝐵𝑊𝜓) ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 (𝐵𝑊𝜓)))
1510, 14bitr4i 281 1 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 (𝐵𝑊𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  {cab 2800  wrex 3131  {crab 3134  Vcvv 3469   cuni 4813  cmpt 5122  ran crn 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-fv 6342
This theorem is referenced by: (None)
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