Theorem relfld 6241

Description: The double union of a relation is its field. (Contributed by NM, 17-Sep-2006.)

Assertion

Ref	Expression
relfld	⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))

Proof of Theorem relfld

Step	Hyp	Ref	Expression
1		relssdmrn 6235	. . . 4 ⊢ (Rel 𝑅 → 𝑅 ⊆ (dom 𝑅 × ran 𝑅))
2		uniss 4873	. . . 4 ⊢ (𝑅 ⊆ (dom 𝑅 × ran 𝑅) → ∪ 𝑅 ⊆ ∪ (dom 𝑅 × ran 𝑅))
3		uniss 4873	. . . 4 ⊢ (∪ 𝑅 ⊆ ∪ (dom 𝑅 × ran 𝑅) → ∪ ∪ 𝑅 ⊆ ∪ ∪ (dom 𝑅 × ran 𝑅))
4	1, 2, 3	3syl 18	. . 3 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 ⊆ ∪ ∪ (dom 𝑅 × ran 𝑅))
5		unixpss 5767	. . 3 ⊢ ∪ ∪ (dom 𝑅 × ran 𝑅) ⊆ (dom 𝑅 ∪ ran 𝑅)
6	4, 5	sstrdi 3948	. 2 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅))
7		dmrnssfld 5931	. . 3 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
8	7	a1i 11	. 2 ⊢ (Rel 𝑅 → (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅)
9	6, 8	eqssd 3953	1 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅))

Syntax hints:   → wi 4   = wceq 1542   ∪ cun 3901   ⊆ wss 3903  ∪ cuni 4865   × cxp 5630  dom cdm 5632  ran crn 5633  Rel wrel 5637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643

This theorem is referenced by:  relresfld  6242  unidmrn  6245  relcnvfld  6246  unixp  6248  relexp0  14958  relexpfld  14984  rtrclreclem4  14996  dfrtrcl2  14997  lefld  18527  fvmptiunrelexplb0da  44038