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Theorem rabfmpunirn 30018
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 3099 . . . . 5 {𝑦𝑊𝜑} = {𝑦 ∣ (𝑦𝑊𝜑)}
32mpteq2i 4976 . . . 4 (𝑥𝑉 ↦ {𝑦𝑊𝜑}) = (𝑥𝑉 ↦ {𝑦 ∣ (𝑦𝑊𝜑)})
41, 3eqtri 2802 . . 3 𝐹 = (𝑥𝑉 ↦ {𝑦 ∣ (𝑦𝑊𝜑)})
5 rabfmpunirn.2 . . . 4 𝑊 ∈ V
65zfausab 5047 . . 3 {𝑦 ∣ (𝑦𝑊𝜑)} ∈ V
7 eleq1 2847 . . . 4 (𝑦 = 𝐵 → (𝑦𝑊𝐵𝑊))
8 rabfmpunirn.3 . . . 4 (𝑦 = 𝐵 → (𝜑𝜓))
97, 8anbi12d 624 . . 3 (𝑦 = 𝐵 → ((𝑦𝑊𝜑) ↔ (𝐵𝑊𝜓)))
104, 6, 9abfmpunirn 30017 . 2 (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 (𝐵𝑊𝜓)))
11 elex 3414 . . . . 5 (𝐵𝑊𝐵 ∈ V)
1211adantr 474 . . . 4 ((𝐵𝑊𝜓) → 𝐵 ∈ V)
1312rexlimivw 3211 . . 3 (∃𝑥𝑉 (𝐵𝑊𝜓) → 𝐵 ∈ V)
1413pm4.71ri 556 . 2 (∃𝑥𝑉 (𝐵𝑊𝜓) ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 (𝐵𝑊𝜓)))
1510, 14bitr4i 270 1 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 (𝐵𝑊𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  {cab 2763  wrex 3091  {crab 3094  Vcvv 3398   cuni 4671  cmpt 4965  ran crn 5356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-fv 6143
This theorem is referenced by: (None)
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