Theorem rabfmpunirn 32794

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 3405	. . . . 5 ⊢ {𝑦 ∈ 𝑊 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)}
3	2	mpteq2i 5186	. . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑊 ∣ 𝜑}) = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)})
4	1, 3	eqtri 2775	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)})
5		rabfmpunirn.2	. . . 4 ⊢ 𝑊 ∈ V
6	5	zfausab 5278	. . 3 ⊢ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)} ∈ V
7		eleq1 2840	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
8		rabfmpunirn.3	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 640	. . 3 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ 𝑊 ∧ 𝜑) ↔ (𝐵 ∈ 𝑊 ∧ 𝜓)))
10	4, 6, 9	abfmpunirn 32793	. 2 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓)))
11		elex 3465	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
12	11	adantr 483	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝜓) → 𝐵 ∈ V)
13	12	rexlimivw 3149	. . 3 ⊢ (∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓) → 𝐵 ∈ V)
14	13	pm4.71ri 567	. 2 ⊢ (∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓) ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓)))
15	10, 14	bitr4i 280	1 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1550   ∈ wcel 2132  {cab 2730  ∃wrex 3076  {crab 3404  Vcvv 3444  ∪ cuni 4855   ↦ cmpt 5171  ran crn 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-10 2165  ax-11 2181  ax-12 2202  ax-ext 2724  ax-sep 5236  ax-nul 5246  ax-pr 5380

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1553  df-fal 1563  df-ex 1790  df-nf 1794  df-sb 2081  df-mo 2556  df-eu 2586  df-clab 2731  df-cleq 2744  df-clel 2827  df-nfc 2901  df-ne 2948  df-ral 3067  df-rex 3077  df-rab 3405  df-v 3446  df-sbc 3736  df-csb 3844  df-dif 3898  df-un 3900  df-in 3902  df-ss 3912  df-nul 4277  df-if 4471  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4856  df-br 5091  df-opab 5153  df-mpt 5172  df-id 5531  df-xp 5642  df-rel 5643  df-cnv 5644  df-co 5645  df-dm 5646  df-rn 5647  df-res 5648  df-ima 5649  df-iota 6462  df-fun 6508  df-fn 6509  df-fv 6514

This theorem is referenced by: (None)