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Theorem rabfmpunirn 29762
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 3059 . . . . 5 {𝑦𝑊𝜑} = {𝑦 ∣ (𝑦𝑊𝜑)}
32mpteq2i 4893 . . . 4 (𝑥𝑉 ↦ {𝑦𝑊𝜑}) = (𝑥𝑉 ↦ {𝑦 ∣ (𝑦𝑊𝜑)})
41, 3eqtri 2782 . . 3 𝐹 = (𝑥𝑉 ↦ {𝑦 ∣ (𝑦𝑊𝜑)})
5 rabfmpunirn.2 . . . 4 𝑊 ∈ V
65zfausab 4962 . . 3 {𝑦 ∣ (𝑦𝑊𝜑)} ∈ V
7 eleq1 2827 . . . 4 (𝑦 = 𝐵 → (𝑦𝑊𝐵𝑊))
8 rabfmpunirn.3 . . . 4 (𝑦 = 𝐵 → (𝜑𝜓))
97, 8anbi12d 749 . . 3 (𝑦 = 𝐵 → ((𝑦𝑊𝜑) ↔ (𝐵𝑊𝜓)))
104, 6, 9abfmpunirn 29761 . 2 (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 (𝐵𝑊𝜓)))
11 elex 3352 . . . . 5 (𝐵𝑊𝐵 ∈ V)
1211adantr 472 . . . 4 ((𝐵𝑊𝜓) → 𝐵 ∈ V)
1312rexlimivw 3167 . . 3 (∃𝑥𝑉 (𝐵𝑊𝜓) → 𝐵 ∈ V)
1413pm4.71ri 668 . 2 (∃𝑥𝑉 (𝐵𝑊𝜓) ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 (𝐵𝑊𝜓)))
1510, 14bitr4i 267 1 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 (𝐵𝑊𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  {crab 3054  Vcvv 3340   cuni 4588  cmpt 4881  ran crn 5267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by: (None)
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