Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rexunirn Structured version   Visualization version   GIF version

Theorem rexunirn 29664
Description: Restricted existential quantification over the union of the range of a function. Cf. rexrn 6504 and eluni2 4576. (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 3066 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
2 19.42v 2032 . . . . 5 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
3 df-rex 3066 . . . . . 6 (∃𝑦𝐵 𝜑 ↔ ∃𝑦(𝑦𝐵𝜑))
43anbi2i 601 . . . . 5 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
52, 4bitr4i 267 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
65exbii 1923 . . 3 (∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
71, 6bitr4i 267 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
8 rexunirn.2 . . . . . . . 8 (𝑥𝐴𝐵𝑉)
9 rexunirn.1 . . . . . . . . 9 𝐹 = (𝑥𝐴𝐵)
109elrnmpt1 5512 . . . . . . . 8 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
118, 10mpdan 659 . . . . . . 7 (𝑥𝐴𝐵 ∈ ran 𝐹)
12 eleq2 2838 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
1312anbi1d 607 . . . . . . . 8 (𝑏 = 𝐵 → ((𝑦𝑏𝜑) ↔ (𝑦𝐵𝜑)))
1413rspcev 3458 . . . . . . 7 ((𝐵 ∈ ran 𝐹 ∧ (𝑦𝐵𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑))
1511, 14sylan 561 . . . . . 6 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑))
16 r19.41v 3236 . . . . . 6 (∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
1715, 16sylib 208 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
1817eximi 1909 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
19 df-rex 3066 . . . . 5 (∃𝑦 ran 𝐹𝜑 ↔ ∃𝑦(𝑦 ran 𝐹𝜑))
20 eluni2 4576 . . . . . . 7 (𝑦 ran 𝐹 ↔ ∃𝑏 ∈ ran 𝐹 𝑦𝑏)
2120anbi1i 602 . . . . . 6 ((𝑦 ran 𝐹𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2221exbii 1923 . . . . 5 (∃𝑦(𝑦 ran 𝐹𝜑) ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2319, 22bitri 264 . . . 4 (∃𝑦 ran 𝐹𝜑 ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2418, 23sylibr 224 . . 3 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦 ran 𝐹𝜑)
2524exlimiv 2009 . 2 (∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦 ran 𝐹𝜑)
267, 25sylbi 207 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦 ran 𝐹𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wex 1851  wcel 2144  wrex 3061   cuni 4572  cmpt 4861  ran crn 5250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-cnv 5257  df-dm 5259  df-rn 5260
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator