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Theorem iunrdx 30249
 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 6589 . . . . . . 7 (𝐹:𝐴onto𝐶𝐹:𝐴𝐶)
31, 2syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐶)
43ffvelrnda 6849 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐶)
5 foelrn 6870 . . . . . 6 ((𝐹:𝐴onto𝐶𝑦𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
61, 5sylan 580 . . . . 5 ((𝜑𝑦𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
7 iunrdx.2 . . . . . 6 ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)
87eleq2d 2903 . . . . 5 ((𝜑𝑦 = (𝐹𝑥)) → (𝑧𝐷𝑧𝐵))
94, 6, 8rexxfrd 5306 . . . 4 (𝜑 → (∃𝑦𝐶 𝑧𝐷 ↔ ∃𝑥𝐴 𝑧𝐵))
109bicomd 224 . . 3 (𝜑 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐶 𝑧𝐷))
1110abbidv 2890 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐶 𝑧𝐷})
12 df-iun 4919 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
13 df-iun 4919 . 2 𝑦𝐶 𝐷 = {𝑧 ∣ ∃𝑦𝐶 𝑧𝐷}
1411, 12, 133eqtr4g 2886 1 (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1530   ∈ wcel 2107  {cab 2804  ∃wrex 3144  ∪ ciun 4917  ⟶wf 6350  –onto→wfo 6352  ‘cfv 6354 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362 This theorem is referenced by:  volmeas  31395
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