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Theorem imaiun1 37459
Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)
Assertion
Ref Expression
imaiun1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem imaiun1
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 3214 . . . 4 (∃𝑥𝐴𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑧𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
2 vex 3192 . . . . . 6 𝑦 ∈ V
32elima3 5437 . . . . 5 (𝑦 ∈ (𝐵𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
43rexbii 3035 . . . 4 (∃𝑥𝐴 𝑦 ∈ (𝐵𝐶) ↔ ∃𝑥𝐴𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
5 eliun 4495 . . . . . . 7 (⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵 ↔ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵)
65anbi2i 729 . . . . . 6 ((𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ (𝑧𝐶 ∧ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵))
7 r19.42v 3085 . . . . . 6 (∃𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑧𝐶 ∧ ∃𝑥𝐴𝑧, 𝑦⟩ ∈ 𝐵))
86, 7bitr4i 267 . . . . 5 ((𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
98exbii 1771 . . . 4 (∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑧𝑥𝐴 (𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
101, 4, 93bitr4ri 293 . . 3 (∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
112elima3 5437 . . 3 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ ∃𝑧(𝑧𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝑥𝐴 𝐵))
12 eliun 4495 . . 3 (𝑦 𝑥𝐴 (𝐵𝐶) ↔ ∃𝑥𝐴 𝑦 ∈ (𝐵𝐶))
1310, 11, 123bitr4i 292 . 2 (𝑦 ∈ ( 𝑥𝐴 𝐵𝐶) ↔ 𝑦 𝑥𝐴 (𝐵𝐶))
1413eqriv 2618 1 ( 𝑥𝐴 𝐵𝐶) = 𝑥𝐴 (𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  cop 4159   ciun 4490  cima 5082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-iun 4492  df-br 4619  df-opab 4679  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092
This theorem is referenced by:  trclimalb2  37534
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