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Theorem iunrdx 29720
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 6257 . . . . . . 7 (𝐹:𝐴onto𝐶𝐹:𝐴𝐶)
31, 2syl 17 . . . . . 6 (𝜑𝐹:𝐴𝐶)
43ffvelrnda 6504 . . . . 5 ((𝜑𝑥𝐴) → (𝐹𝑥) ∈ 𝐶)
5 foelrn 6523 . . . . . 6 ((𝐹:𝐴onto𝐶𝑦𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
61, 5sylan 569 . . . . 5 ((𝜑𝑦𝐶) → ∃𝑥𝐴 𝑦 = (𝐹𝑥))
7 iunrdx.2 . . . . . 6 ((𝜑𝑦 = (𝐹𝑥)) → 𝐷 = 𝐵)
87eleq2d 2836 . . . . 5 ((𝜑𝑦 = (𝐹𝑥)) → (𝑧𝐷𝑧𝐵))
94, 6, 8rexxfrd 5010 . . . 4 (𝜑 → (∃𝑦𝐶 𝑧𝐷 ↔ ∃𝑥𝐴 𝑧𝐵))
109bicomd 213 . . 3 (𝜑 → (∃𝑥𝐴 𝑧𝐵 ↔ ∃𝑦𝐶 𝑧𝐷))
1110abbidv 2890 . 2 (𝜑 → {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵} = {𝑧 ∣ ∃𝑦𝐶 𝑧𝐷})
12 df-iun 4657 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∃𝑥𝐴 𝑧𝐵}
13 df-iun 4657 . 2 𝑦𝐶 𝐷 = {𝑧 ∣ ∃𝑦𝐶 𝑧𝐷}
1411, 12, 133eqtr4g 2830 1 (𝜑 𝑥𝐴 𝐵 = 𝑦𝐶 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  {cab 2757  wrex 3062   ciun 4655  wf 6026  ontowfo 6028  cfv 6030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fo 6036  df-fv 6038
This theorem is referenced by:  volmeas  30634
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