Theorem rabfmpunirn 30990

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 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑊 ∣ 𝜑})
rabfmpunirn.2	⊢ 𝑊 ∈ V
rabfmpunirn.3	⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rabfmpunirn	⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓))

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

Step	Hyp	Ref	Expression
1		rabfmpunirn.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑊 ∣ 𝜑})
2		df-rab 3073	. . . . 5 ⊢ {𝑦 ∈ 𝑊 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)}
3	2	mpteq2i 5179	. . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑊 ∣ 𝜑}) = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)})
4	1, 3	eqtri 2766	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)})
5		rabfmpunirn.2	. . . 4 ⊢ 𝑊 ∈ V
6	5	zfausab 5254	. . 3 ⊢ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)} ∈ V
7		eleq1 2826	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
8		rabfmpunirn.3	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 631	. . 3 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ 𝑊 ∧ 𝜑) ↔ (𝐵 ∈ 𝑊 ∧ 𝜓)))
10	4, 6, 9	abfmpunirn 30989	. 2 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓)))
11		elex 3450	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
12	11	adantr 481	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝜓) → 𝐵 ∈ V)
13	12	rexlimivw 3211	. . 3 ⊢ (∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓) → 𝐵 ∈ V)
14	13	pm4.71ri 561	. 2 ⊢ (∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓) ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓)))
15	10, 14	bitr4i 277	1 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {cab 2715  ∃wrex 3065  {crab 3068  Vcvv 3432  ∪ cuni 4839   ↦ cmpt 5157  ran crn 5590

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by: (None)