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Theorem rexunirn 30270
 Description: Restricted existential quantification over the union of the range of a function. Cf. rexrn 6830 and eluni2 4804. (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 3112 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
2 19.42v 1954 . . . . 5 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
3 df-rex 3112 . . . . . 6 (∃𝑦𝐵 𝜑 ↔ ∃𝑦(𝑦𝐵𝜑))
43anbi2i 625 . . . . 5 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
52, 4bitr4i 281 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
65exbii 1849 . . 3 (∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
71, 6bitr4i 281 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
8 rexunirn.2 . . . . . . . 8 (𝑥𝐴𝐵𝑉)
9 rexunirn.1 . . . . . . . . 9 𝐹 = (𝑥𝐴𝐵)
109elrnmpt1 5794 . . . . . . . 8 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
118, 10mpdan 686 . . . . . . 7 (𝑥𝐴𝐵 ∈ ran 𝐹)
12 eleq2 2878 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
1312anbi1d 632 . . . . . . . 8 (𝑏 = 𝐵 → ((𝑦𝑏𝜑) ↔ (𝑦𝐵𝜑)))
1413rspcev 3571 . . . . . . 7 ((𝐵 ∈ ran 𝐹 ∧ (𝑦𝐵𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑))
1511, 14sylan 583 . . . . . 6 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑))
16 r19.41v 3300 . . . . . 6 (∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
1715, 16sylib 221 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
1817eximi 1836 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
19 df-rex 3112 . . . . 5 (∃𝑦 ran 𝐹𝜑 ↔ ∃𝑦(𝑦 ran 𝐹𝜑))
20 eluni2 4804 . . . . . . 7 (𝑦 ran 𝐹 ↔ ∃𝑏 ∈ ran 𝐹 𝑦𝑏)
2120anbi1i 626 . . . . . 6 ((𝑦 ran 𝐹𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2221exbii 1849 . . . . 5 (∃𝑦(𝑦 ran 𝐹𝜑) ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2319, 22bitri 278 . . . 4 (∃𝑦 ran 𝐹𝜑 ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2418, 23sylibr 237 . . 3 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦 ran 𝐹𝜑)
2524exlimiv 1931 . 2 (∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦 ran 𝐹𝜑)
267, 25sylbi 220 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦 ran 𝐹𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  ∪ cuni 4800   ↦ cmpt 5110  ran crn 5520 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-cnv 5527  df-dm 5529  df-rn 5530 This theorem is referenced by: (None)
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