Theorem dfiun3g 5964

Description: Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.)

Assertion

Ref	Expression
dfiun3g	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵))

Proof of Theorem dfiun3g

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiun2g 5034	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
2		eqid 2733	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	rnmpt 5955	. . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
4	3	unieqi 4922	. 2 ⊢ ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
5	1, 4	eqtr4di 2791	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062  ∃wrex 3071  ∪ cuni 4909  ∪ ciun 4998   ↦ cmpt 5232  ran crn 5678

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 5300  ax-nul 5307  ax-pr 5428

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-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-cnv 5685  df-dm 5687  df-rn 5688

This theorem is referenced by:  dfiun3  5966  iunon  8339  onoviun  8343  gruiun  10794  tgiun  22482  acunirnmpt2f  31886  locfinreflem  32820  carsgclctunlem2  33318  pmeasadd  33324  saliunclf  45038  salexct3  45058  salgensscntex  45060  meadjiun  45182  omeiunle  45233  ovolval5lem2  45369