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Theorem iunrdx 29851
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 6300 . . . . . . 7 (𝐹:𝐴onto𝐶𝐹:𝐴𝐶)
31, 2syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐶)
43ffvelrnda 6553 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐶)
5 foelrn 6572 . . . . . 6 ((𝐹:𝐴onto𝐶𝑦𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
61, 5sylan 575 . . . . 5 ((𝜑𝑦𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
7 iunrdx.2 . . . . . 6 ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)
87eleq2d 2830 . . . . 5 ((𝜑𝑦 = (𝐹𝑥)) → (𝑧𝐷𝑧𝐵))
94, 6, 8rexxfrd 5046 . . . 4 (𝜑 → (∃𝑦𝐶 𝑧𝐷 ↔ ∃𝑥𝐴 𝑧𝐵))
109bicomd 214 . . 3 (𝜑 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐶 𝑧𝐷))
1110abbidv 2884 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐶 𝑧𝐷})
12 df-iun 4680 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
13 df-iun 4680 . 2 𝑦𝐶 𝐷 = {𝑧 ∣ ∃𝑦𝐶 𝑧𝐷}
1411, 12, 133eqtr4g 2824 1 (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  {cab 2751  wrex 3056   ciun 4678  wf 6066  ontowfo 6068  cfv 6070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fo 6076  df-fv 6078
This theorem is referenced by:  volmeas  30762
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