Theorem rexunirn 33130

Description: Restricted existential quantification over the union of the range of a function. Cf. rexrn 7087 and eluni2 4870. (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 3087	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
2		19.42v 1985	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)))
3		df-rex 3087	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))
4	3	anbi2i 635	. . . . 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 1880	. . 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 5941	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)
11	8, 10	mpdan 700	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ ran 𝐹)
12		eleq2 2849	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝐵))
13	12	anbi1d 643	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝑦 ∈ 𝑏 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜑)))
14	13	rspcev 3576	. . . . . . 7 ⊢ ((𝐵 ∈ ran 𝐹 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦 ∈ 𝑏 ∧ 𝜑))
15	11, 14	sylan 592	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦 ∈ 𝑏 ∧ 𝜑))
16		r19.41v 3192	. . . . . 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 1867	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
19		df-rex 3087	. . . . 5 ⊢ (∃𝑦 ∈ ∪ ran 𝐹𝜑 ↔ ∃𝑦(𝑦 ∈ ∪ ran 𝐹 ∧ 𝜑))
20		eluni2 4870	. . . . . . 7 ⊢ (𝑦 ∈ ∪ ran 𝐹 ↔ ∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏)
21	20	anbi1i 636	. . . . . 6 ⊢ ((𝑦 ∈ ∪ ran 𝐹 ∧ 𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
22	21	exbii 1880	. . . . 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 1962	. 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)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∃wrex 3086  ∪ cuni 4866   ↦ cmpt 5185  ran crn 5651

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-cnv 5658  df-dm 5660  df-rn 5661

This theorem is used by: (None)