Theorem rabfmpunirn 33128

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 3415	. . . . 5 ⊢ {𝑦 ∈ 𝑊 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)}
3	2	mpteq2i 5205	. . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑊 ∣ 𝜑}) = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)})
4	1, 3	eqtri 2785	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)})
5		rabfmpunirn.2	. . . 4 ⊢ 𝑊 ∈ V
6		sepab 5301	. . . 4 ⊢ (𝑊 ∈ V → {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)} ∈ V)
7	5, 6	ax-mp 5	. . 3 ⊢ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)} ∈ V
8		eleq1 2850	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
9		rabfmpunirn.3	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))
10	8, 9	anbi12d 644	. . 3 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ 𝑊 ∧ 𝜑) ↔ (𝐵 ∈ 𝑊 ∧ 𝜓)))
11	4, 7, 10	abfmpunirn 33127	. 2 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓)))
12		elex 3474	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
13	12	adantr 486	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝜓) → 𝐵 ∈ V)
14	13	rexlimivw 3161	. . 3 ⊢ (∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓) → 𝐵 ∈ V)
15	14	pm4.71ri 570	. 2 ⊢ (∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓) ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓)))
16	11, 15	bitr4i 281	1 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2740  ∃wrex 3088  {crab 3414  Vcvv 3453  ∪ cuni 4870   ↦ cmpt 5190  ran crn 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545

This theorem is used by: (None)