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Theorem iunrdx 32062
Description: Re-index an indexed union. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Hypotheses
Ref Expression
iunrdx.1 (𝜑𝐹:𝐴onto𝐶)
iunrdx.2 ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)
Assertion
Ref Expression
iunrdx (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝑥,𝐶,𝑦   𝑥,𝐷   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑦)

Proof of Theorem iunrdx
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 iunrdx.1 . . . . . . 7 (𝜑𝐹:𝐴onto𝐶)
2 fof 6804 . . . . . . 7 (𝐹:𝐴onto𝐶𝐹:𝐴𝐶)
31, 2syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐶)
43ffvelcdmda 7085 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐶)
5 foelrn 7107 . . . . . 6 ((𝐹:𝐴onto𝐶𝑦𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
61, 5sylan 578 . . . . 5 ((𝜑𝑦𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
7 iunrdx.2 . . . . . 6 ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)
87eleq2d 2817 . . . . 5 ((𝜑𝑦 = (𝐹𝑥)) → (𝑧𝐷𝑧𝐵))
94, 6, 8rexxfrd 5406 . . . 4 (𝜑 → (∃𝑦𝐶 𝑧𝐷 ↔ ∃𝑥𝐴 𝑧𝐵))
109bicomd 222 . . 3 (𝜑 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐶 𝑧𝐷))
1110abbidv 2799 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐶 𝑧𝐷})
12 df-iun 4998 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
13 df-iun 4998 . 2 𝑦𝐶 𝐷 = {𝑧 ∣ ∃𝑦𝐶 𝑧𝐷}
1411, 12, 133eqtr4g 2795 1 (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  {cab 2707  wrex 3068   ciun 4996  wf 6538  ontowfo 6540  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550
This theorem is referenced by:  volmeas  33527
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