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Theorem rexunirn 30037
Description: Restricted existential quantification over the union of the range of a function. Cf. rexrn 6678 and eluni2 4716. (Contributed by Thierry Arnoux, 19-Sep-2017.)
Hypotheses
Ref Expression
rexunirn.1 𝐹 = (𝑥𝐴𝐵)
rexunirn.2 (𝑥𝐴𝐵𝑉)
Assertion
Ref Expression
rexunirn (∃𝑥𝐴𝑦𝐵 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.
StepHypRef Expression
1 df-rex 3094 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
2 19.42v 1912 . . . . 5 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
3 df-rex 3094 . . . . . 6 (∃𝑦𝐵 𝜑 ↔ ∃𝑦(𝑦𝐵𝜑))
43anbi2i 613 . . . . 5 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
52, 4bitr4i 270 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
65exbii 1810 . . 3 (∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
71, 6bitr4i 270 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
8 rexunirn.2 . . . . . . . 8 (𝑥𝐴𝐵𝑉)
9 rexunirn.1 . . . . . . . . 9 𝐹 = (𝑥𝐴𝐵)
109elrnmpt1 5673 . . . . . . . 8 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
118, 10mpdan 674 . . . . . . 7 (𝑥𝐴𝐵 ∈ ran 𝐹)
12 eleq2 2854 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
1312anbi1d 620 . . . . . . . 8 (𝑏 = 𝐵 → ((𝑦𝑏𝜑) ↔ (𝑦𝐵𝜑)))
1413rspcev 3535 . . . . . . 7 ((𝐵 ∈ ran 𝐹 ∧ (𝑦𝐵𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑))
1511, 14sylan 572 . . . . . 6 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑))
16 r19.41v 3288 . . . . . 6 (∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
1715, 16sylib 210 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
1817eximi 1797 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
19 df-rex 3094 . . . . 5 (∃𝑦 ran 𝐹𝜑 ↔ ∃𝑦(𝑦 ran 𝐹𝜑))
20 eluni2 4716 . . . . . . 7 (𝑦 ran 𝐹 ↔ ∃𝑏 ∈ ran 𝐹 𝑦𝑏)
2120anbi1i 614 . . . . . 6 ((𝑦 ran 𝐹𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2221exbii 1810 . . . . 5 (∃𝑦(𝑦 ran 𝐹𝜑) ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2319, 22bitri 267 . . . 4 (∃𝑦 ran 𝐹𝜑 ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2418, 23sylibr 226 . . 3 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦 ran 𝐹𝜑)
2524exlimiv 1889 . 2 (∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦 ran 𝐹𝜑)
267, 25sylbi 209 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦 ran 𝐹𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wex 1742  wcel 2050  wrex 3089   cuni 4712  cmpt 5008  ran crn 5408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-cnv 5415  df-dm 5417  df-rn 5418
This theorem is referenced by: (None)
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