Theorem rexunirn 30559

Description: Restricted existential quantification over the union of the range of a function. Cf. rexrn 6906 and eluni2 4823. (Contributed by Thierry Arnoux, 19-Sep-2017.)

Hypotheses

Ref	Expression
rexunirn.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
rexunirn.2	⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉)

Assertion

Ref	Expression
rexunirn	⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ ∪ ran 𝐹𝜑)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴   𝑥,𝐹   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐹(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem rexunirn

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 3067	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
2		19.42v 1962	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)))
3		df-rex 3067	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))
4	3	anbi2i 626	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)))
5	2, 4	bitr4i 281	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
6	5	exbii 1855	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
7	1, 6	bitr4i 281	. 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)))
8		rexunirn.2	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉)
9		rexunirn.1	. . . . . . . . 9 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
10	9	elrnmpt1 5827	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)
11	8, 10	mpdan 687	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ ran 𝐹)
12		eleq2 2826	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝐵))
13	12	anbi1d 633	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝑦 ∈ 𝑏 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜑)))
14	13	rspcev 3537	. . . . . . 7 ⊢ ((𝐵 ∈ ran 𝐹 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦 ∈ 𝑏 ∧ 𝜑))
15	11, 14	sylan 583	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦 ∈ 𝑏 ∧ 𝜑))
16		r19.41v 3260	. . . . . 6 ⊢ (∃𝑏 ∈ ran 𝐹(𝑦 ∈ 𝑏 ∧ 𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
17	15, 16	sylib 221	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → (∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
18	17	eximi 1842	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
19		df-rex 3067	. . . . 5 ⊢ (∃𝑦 ∈ ∪ ran 𝐹𝜑 ↔ ∃𝑦(𝑦 ∈ ∪ ran 𝐹 ∧ 𝜑))
20		eluni2 4823	. . . . . . 7 ⊢ (𝑦 ∈ ∪ ran 𝐹 ↔ ∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏)
21	20	anbi1i 627	. . . . . 6 ⊢ ((𝑦 ∈ ∪ ran 𝐹 ∧ 𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
22	21	exbii 1855	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ ∪ ran 𝐹 ∧ 𝜑) ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
23	19, 22	bitri 278	. . . 4 ⊢ (∃𝑦 ∈ ∪ ran 𝐹𝜑 ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
24	18, 23	sylibr 237	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑦 ∈ ∪ ran 𝐹𝜑)
25	24	exlimiv 1938	. 2 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑦 ∈ ∪ ran 𝐹𝜑)
26	7, 25	sylbi 220	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ ∪ ran 𝐹𝜑)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2110  ∃wrex 3062  ∪ cuni 4819   ↦ cmpt 5135  ran crn 5552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-cnv 5559  df-dm 5561  df-rn 5562

This theorem is referenced by: (None)