Theorem abfmpunirn 30891

Description: Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)

Hypotheses

Ref	Expression
abfmpunirn.1	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑})
abfmpunirn.2	⊢ {𝑦 ∣ 𝜑} ∈ V
abfmpunirn.3	⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)

Proof of Theorem abfmpunirn

Step	Hyp	Ref	Expression
1		elex 3440	. 2 ⊢ (𝐵 ∈ ∪ ran 𝐹 → 𝐵 ∈ V)
2		abfmpunirn.2	. . . . . 6 ⊢ {𝑦 ∣ 𝜑} ∈ V
3		abfmpunirn.1	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ 𝜑})
4	2, 3	fnmpti 6560	. . . . 5 ⊢ 𝐹 Fn 𝑉
5		fnunirn 7108	. . . . 5 ⊢ (𝐹 Fn 𝑉 → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ (𝐹‘𝑥)))
6	4, 5	ax-mp 5	. . . 4 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ (𝐹‘𝑥))
7	3	fvmpt2 6868	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ {𝑦 ∣ 𝜑} ∈ V) → (𝐹‘𝑥) = {𝑦 ∣ 𝜑})
8	2, 7	mpan2 687	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → (𝐹‘𝑥) = {𝑦 ∣ 𝜑})
9	8	eleq2d 2824	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → (𝐵 ∈ (𝐹‘𝑥) ↔ 𝐵 ∈ {𝑦 ∣ 𝜑}))
10	9	rexbiia 3176	. . . 4 ⊢ (∃𝑥 ∈ 𝑉 𝐵 ∈ (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑})
11	6, 10	bitri 274	. . 3 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑})
12		abfmpunirn.3	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))
13	12	elabg 3600	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓))
14	13	rexbidv 3225	. . 3 ⊢ (𝐵 ∈ V → (∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝑉 𝜓))
15	11, 14	syl5bb 282	. 2 ⊢ (𝐵 ∈ V → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝜓))
16	1, 15	biadanii 818	1 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {cab 2715  ∃wrex 3064  Vcvv 3422  ∪ cuni 4836   ↦ cmpt 5153  ran crn 5581   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by:  rabfmpunirn  30892  isrnsiga  31981  isrnmeas  32068