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Theorem rexunirn 29303
Description: Restricted existential quantification over the union of the range of a function. Cf. rexrn 6347 and eluni2 4431. (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 2915 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
2 19.42v 1916 . . . . 5 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
3 df-rex 2915 . . . . . 6 (∃𝑦𝐵 𝜑 ↔ ∃𝑦(𝑦𝐵𝜑))
43anbi2i 729 . . . . 5 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
52, 4bitr4i 267 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
65exbii 1772 . . 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 5363 . . . . . . . 8 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
118, 10mpdan 701 . . . . . . 7 (𝑥𝐴𝐵 ∈ ran 𝐹)
12 eleq2 2688 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
1312anbi1d 740 . . . . . . . 8 (𝑏 = 𝐵 → ((𝑦𝑏𝜑) ↔ (𝑦𝐵𝜑)))
1413rspcev 3304 . . . . . . 7 ((𝐵 ∈ ran 𝐹 ∧ (𝑦𝐵𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑))
1511, 14sylan 488 . . . . . 6 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑))
16 r19.41v 3084 . . . . . 6 (∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
1715, 16sylib 208 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
1817eximi 1760 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
19 df-rex 2915 . . . . 5 (∃𝑦 ran 𝐹𝜑 ↔ ∃𝑦(𝑦 ran 𝐹𝜑))
20 eluni2 4431 . . . . . . 7 (𝑦 ran 𝐹 ↔ ∃𝑏 ∈ ran 𝐹 𝑦𝑏)
2120anbi1i 730 . . . . . 6 ((𝑦 ran 𝐹𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2221exbii 1772 . . . . 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 1856 . 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 384   = wceq 1481  wex 1702  wcel 1988  wrex 2910   cuni 4427  cmpt 4720  ran crn 5105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-cnv 5112  df-dm 5114  df-rn 5115
This theorem is referenced by: (None)
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