Theorem rexunirn 32583

Description: Restricted existential quantification over the union of the range of a function. Cf. rexrn 7032 and eluni2 4845. (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 3066	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
2		19.42v 1961	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)))
3		df-rex 3066	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))
4	3	anbi2i 630	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)))
5	2, 4	bitr4i 280	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
6	5	exbii 1856	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
7	1, 6	bitr4i 280	. 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 5909	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)
11	8, 10	mpdan 694	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ ran 𝐹)
12		eleq2 2830	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝐵))
13	12	anbi1d 638	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝑦 ∈ 𝑏 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜑)))
14	13	rspcev 3562	. . . . . . 7 ⊢ ((𝐵 ∈ ran 𝐹 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦 ∈ 𝑏 ∧ 𝜑))
15	11, 14	sylan 587	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦 ∈ 𝑏 ∧ 𝜑))
16		r19.41v 3171	. . . . . 6 ⊢ (∃𝑏 ∈ ran 𝐹(𝑦 ∈ 𝑏 ∧ 𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
17	15, 16	sylib 220	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → (∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
18	17	eximi 1843	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
19		df-rex 3066	. . . . 5 ⊢ (∃𝑦 ∈ ∪ ran 𝐹𝜑 ↔ ∃𝑦(𝑦 ∈ ∪ ran 𝐹 ∧ 𝜑))
20		eluni2 4845	. . . . . . 7 ⊢ (𝑦 ∈ ∪ ran 𝐹 ↔ ∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏)
21	20	anbi1i 631	. . . . . 6 ⊢ ((𝑦 ∈ ∪ ran 𝐹 ∧ 𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
22	21	exbii 1856	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ ∪ ran 𝐹 ∧ 𝜑) ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
23	19, 22	bitri 277	. . . 4 ⊢ (∃𝑦 ∈ ∪ ran 𝐹𝜑 ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
24	18, 23	sylibr 236	. . 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 219	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ ∪ ran 𝐹𝜑)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548  ∃wex 1787   ∈ wcel 2121  ∃wrex 3065  ∪ cuni 4841   ↦ cmpt 5156  ran crn 5622

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-cnv 5629  df-dm 5631  df-rn 5632

This theorem is referenced by: (None)