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Theorem rexunirn 32520
Description: Restricted existential quantification over the union of the range of a function. Cf. rexrn 7107 and eluni2 4916. (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 3069 . . 3 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
2 19.42v 1951 . . . . 5 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
3 df-rex 3069 . . . . . 6 (∃𝑦𝐵 𝜑 ↔ ∃𝑦(𝑦𝐵𝜑))
43anbi2i 623 . . . . 5 ((𝑥𝐴 ∧ ∃𝑦𝐵 𝜑) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦𝐵𝜑)))
52, 4bitr4i 278 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ (𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
65exbii 1845 . . 3 (∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) ↔ ∃𝑥(𝑥𝐴 ∧ ∃𝑦𝐵 𝜑))
71, 6bitr4i 278 . 2 (∃𝑥𝐴𝑦𝐵 𝜑 ↔ ∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)))
8 rexunirn.2 . . . . . . . 8 (𝑥𝐴𝐵𝑉)
9 rexunirn.1 . . . . . . . . 9 𝐹 = (𝑥𝐴𝐵)
109elrnmpt1 5974 . . . . . . . 8 ((𝑥𝐴𝐵𝑉) → 𝐵 ∈ ran 𝐹)
118, 10mpdan 687 . . . . . . 7 (𝑥𝐴𝐵 ∈ ran 𝐹)
12 eleq2 2828 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
1312anbi1d 631 . . . . . . . 8 (𝑏 = 𝐵 → ((𝑦𝑏𝜑) ↔ (𝑦𝐵𝜑)))
1413rspcev 3622 . . . . . . 7 ((𝐵 ∈ ran 𝐹 ∧ (𝑦𝐵𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑))
1511, 14sylan 580 . . . . . 6 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑))
16 r19.41v 3187 . . . . . 6 (∃𝑏 ∈ ran 𝐹(𝑦𝑏𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
1715, 16sylib 218 . . . . 5 ((𝑥𝐴 ∧ (𝑦𝐵𝜑)) → (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
1817eximi 1832 . . . 4 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
19 df-rex 3069 . . . . 5 (∃𝑦 ran 𝐹𝜑 ↔ ∃𝑦(𝑦 ran 𝐹𝜑))
20 eluni2 4916 . . . . . . 7 (𝑦 ran 𝐹 ↔ ∃𝑏 ∈ ran 𝐹 𝑦𝑏)
2120anbi1i 624 . . . . . 6 ((𝑦 ran 𝐹𝜑) ↔ (∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2221exbii 1845 . . . . 5 (∃𝑦(𝑦 ran 𝐹𝜑) ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2319, 22bitri 275 . . . 4 (∃𝑦 ran 𝐹𝜑 ↔ ∃𝑦(∃𝑏 ∈ ran 𝐹 𝑦𝑏𝜑))
2418, 23sylibr 234 . . 3 (∃𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦 ran 𝐹𝜑)
2524exlimiv 1928 . 2 (∃𝑥𝑦(𝑥𝐴 ∧ (𝑦𝐵𝜑)) → ∃𝑦 ran 𝐹𝜑)
267, 25sylbi 217 1 (∃𝑥𝐴𝑦𝐵 𝜑 → ∃𝑦 ran 𝐹𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wrex 3068   cuni 4912  cmpt 5231  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by: (None)
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