Theorem rniun 5193

Description: The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.)

Assertion

Ref	Expression
rniun	⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵

Proof of Theorem rniun

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom4 2845	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦⟨𝑦, 𝑧⟩ ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
2		vex 2824	. . . . . 6 ⊢ 𝑧 ∈ V
3	2	elrn2 5019	. . . . 5 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑦⟨𝑦, 𝑧⟩ ∈ 𝐵)
4	3	rexbii 2557	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦⟨𝑦, 𝑧⟩ ∈ 𝐵)
5		eliun 4011	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
6	5	exbii 1658	. . . 4 ⊢ (∃𝑦⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 ⟨𝑦, 𝑧⟩ ∈ 𝐵)
7	1, 4, 6	3bitr4ri 213	. . 3 ⊢ (∃𝑦⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵)
8	2	elrn2 5019	. . 3 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
9		eliun 4011	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵)
10	7, 8, 9	3bitr4i 212	. 2 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵)
11	10	eqriv 2235	1 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵

Syntax hints:   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∃wrex 2529  ⟨cop 3708  ∪ ciun 4007  ran crn 4770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-iun 4009  df-br 4126  df-opab 4188  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  rnuni  5194  fun11iun  5655  ennnfonelemrn  13288