Theorem rabfmpunirn 30392

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 3147	. . . . 5 ⊢ {𝑦 ∈ 𝑊 ∣ 𝜑} = {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)}
3	2	mpteq2i 5150	. . . 4 ⊢ (𝑥 ∈ 𝑉 ↦ {𝑦 ∈ 𝑊 ∣ 𝜑}) = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)})
4	1, 3	eqtri 2844	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)})
5		rabfmpunirn.2	. . . 4 ⊢ 𝑊 ∈ V
6	5	zfausab 5225	. . 3 ⊢ {𝑦 ∣ (𝑦 ∈ 𝑊 ∧ 𝜑)} ∈ V
7		eleq1 2900	. . . 4 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑊 ↔ 𝐵 ∈ 𝑊))
8		rabfmpunirn.3	. . . 4 ⊢ (𝑦 = 𝐵 → (𝜑 ↔ 𝜓))
9	7, 8	anbi12d 632	. . 3 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ 𝑊 ∧ 𝜑) ↔ (𝐵 ∈ 𝑊 ∧ 𝜓)))
10	4, 6, 9	abfmpunirn 30391	. 2 ⊢ (𝐵 ∈ ∪ ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓)))
11		elex 3512	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
12	11	adantr 483	. . . 4 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝜓) → 𝐵 ∈ V)
13	12	rexlimivw 3282	. . 3 ⊢ (∃𝑥 ∈ 𝑉 (𝐵 ∈ 𝑊 ∧ 𝜓) → 𝐵 ∈ V)
14	13	pm4.71ri 563	. 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 1533   ∈ wcel 2110  {cab 2799  ∃wrex 3139  {crab 3142  Vcvv 3494  ∪ cuni 4831   ↦ cmpt 5138  ran crn 5550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357

This theorem is referenced by: (None)