Theorem abfmpunirn 29452

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 3212	. 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 6022	. . . . 5 ⊢ 𝐹 Fn 𝑉
5		fnunirn 6511	. . . . 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 6291	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ {𝑦 ∣ 𝜑} ∈ V) → (𝐹‘𝑥) = {𝑦 ∣ 𝜑})
8	2, 7	mpan2 707	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → (𝐹‘𝑥) = {𝑦 ∣ 𝜑})
9	8	eleq2d 2687	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → (𝐵 ∈ (𝐹‘𝑥) ↔ 𝐵 ∈ {𝑦 ∣ 𝜑}))
10	9	rexbiia 3040	. . . 4 ⊢ (∃𝑥 ∈ 𝑉 𝐵 ∈ (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑})
11	6, 10	bitri 264	. . 3 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑})
12		abfmpunirn.3	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))
13	12	elabg 3351	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓))
14	13	rexbidv 3052	. . 3 ⊢ (𝐵 ∈ V → (∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝑉 𝜓))
15	11, 14	syl5bb 272	. 2 ⊢ (𝐵 ∈ V → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝜓))
16	1, 15	biadan2 674	1 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 𝜓))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∃wrex 2913  Vcvv 3200  ∪ cuni 4436   ↦ cmpt 4729  ran crn 5115   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  rabfmpunirn  29453  isrnsigaOLD  30175  isrnsiga  30176  isrnmeas  30263