Theorem abfmpunirn 33008

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 3475	. 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 6678	. . . . 5 ⊢ 𝐹 Fn 𝑉
5		fnunirn 7251	. . . . 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 7001	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑉 ∧ {𝑦 ∣ 𝜑} ∈ V) → (𝐹‘𝑥) = {𝑦 ∣ 𝜑})
8	2, 7	mpan2 703	. . . . . 6 ⊢ (𝑥 ∈ 𝑉 → (𝐹‘𝑥) = {𝑦 ∣ 𝜑})
9	8	eleq2d 2848	. . . . 5 ⊢ (𝑥 ∈ 𝑉 → (𝐵 ∈ (𝐹‘𝑥) ↔ 𝐵 ∈ {𝑦 ∣ 𝜑}))
10	9	rexbiia 3109	. . . 4 ⊢ (∃𝑥 ∈ 𝑉 𝐵 ∈ (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑})
11	6, 10	bitri 278	. . 3 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑})
12		abfmpunirn.3	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))
13	12	elabg 3634	. . . 4 ⊢ (𝐵 ∈ V → (𝐵 ∈ {𝑦 ∣ 𝜑} ↔ 𝜓))
14	13	rexbidv 3188	. . 3 ⊢ (𝐵 ∈ V → (∃𝑥 ∈ 𝑉 𝐵 ∈ {𝑦 ∣ 𝜑} ↔ ∃𝑥 ∈ 𝑉 𝜓))
15	11, 14	bitrid 286	. 2 ⊢ (𝐵 ∈ V → (𝐵 ∈ ∪ ran 𝐹 ↔ ∃𝑥 ∈ 𝑉 𝜓))
16	1, 15	biadanii 833	1 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  {cab 2740  ∃wrex 3088  Vcvv 3454  ∪ cuni 4871   ↦ cmpt 5191  ran crn 5661   Fn wfn 6531  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544

This theorem is used by:  rabfmpunirn  33009  isrnsiga  34512  isrnmeas  34599