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Theorem iunrdx 31772
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 6801 . . . . . . 7 (𝐹:𝐴onto𝐶𝐹:𝐴𝐶)
31, 2syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐶)
43ffvelcdmda 7081 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐶)
5 foelrn 7102 . . . . . 6 ((𝐹:𝐴onto𝐶𝑦𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
61, 5sylan 581 . . . . 5 ((𝜑𝑦𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
7 iunrdx.2 . . . . . 6 ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)
87eleq2d 2820 . . . . 5 ((𝜑𝑦 = (𝐹𝑥)) → (𝑧𝐷𝑧𝐵))
94, 6, 8rexxfrd 5405 . . . 4 (𝜑 → (∃𝑦𝐶 𝑧𝐷 ↔ ∃𝑥𝐴 𝑧𝐵))
109bicomd 222 . . 3 (𝜑 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐶 𝑧𝐷))
1110abbidv 2802 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐶 𝑧𝐷})
12 df-iun 4997 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
13 df-iun 4997 . 2 𝑦𝐶 𝐷 = {𝑧 ∣ ∃𝑦𝐶 𝑧𝐷}
1411, 12, 133eqtr4g 2798 1 (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2710  wrex 3071   ciun 4995  wf 6535  ontowfo 6537  cfv 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-fo 6545  df-fv 6547
This theorem is referenced by:  volmeas  33166
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