Theorem rexunirn 32473

Description: Restricted existential quantification over the union of the range of a function. Cf. rexrn 7077 and eluni2 4887. (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 3061	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
2		19.42v 1953	. . . . 5 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)))
3		df-rex 3061	. . . . . 6 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))
4	3	anbi2i 623	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)))
5	2, 4	bitr4i 278	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
6	5	exbii 1848	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ∃𝑦 ∈ 𝐵 𝜑))
7	1, 6	bitr4i 278	. 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 5940	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ ran 𝐹)
11	8, 10	mpdan 687	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐵 ∈ ran 𝐹)
12		eleq2 2823	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝐵))
13	12	anbi1d 631	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → ((𝑦 ∈ 𝑏 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜑)))
14	13	rspcev 3601	. . . . . . 7 ⊢ ((𝐵 ∈ ran 𝐹 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦 ∈ 𝑏 ∧ 𝜑))
15	11, 14	sylan 580	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦 ∈ 𝑏 ∧ 𝜑))
16		r19.41v 3174	. . . . . 6 ⊢ (∃𝑏 ∈ ran 𝐹(𝑦 ∈ 𝑏 ∧ 𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
17	15, 16	sylib 218	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → (∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
18	17	eximi 1835	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
19		df-rex 3061	. . . . 5 ⊢ (∃𝑦 ∈ ∪ ran 𝐹𝜑 ↔ ∃𝑦(𝑦 ∈ ∪ ran 𝐹 ∧ 𝜑))
20		eluni2 4887	. . . . . . 7 ⊢ (𝑦 ∈ ∪ ran 𝐹 ↔ ∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏)
21	20	anbi1i 624	. . . . . 6 ⊢ ((𝑦 ∈ ∪ ran 𝐹 ∧ 𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
22	21	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ ∪ ran 𝐹 ∧ 𝜑) ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
23	19, 22	bitri 275	. . . 4 ⊢ (∃𝑦 ∈ ∪ ran 𝐹𝜑 ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦 ∈ 𝑏 ∧ 𝜑))
24	18, 23	sylibr 234	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑦 ∈ ∪ ran 𝐹𝜑)
25	24	exlimiv 1930	. 2 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)) → ∃𝑦 ∈ ∪ ran 𝐹𝜑)
26	7, 25	sylbi 217	1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ ∪ ran 𝐹𝜑)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3060  ∪ cuni 4883   ↦ cmpt 5201  ran crn 5655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-cnv 5662  df-dm 5664  df-rn 5665

This theorem is referenced by: (None)