Theorem fvssunirnOLD 6915

Description: Obsolete version of fvssunirn 6914 as of 13-Jan-2025. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 31-Aug-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
fvssunirnOLD	⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Proof of Theorem fvssunirnOLD

Step	Hyp	Ref	Expression
1		fvrn0 6911	. . 3 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})
2		elssuni 4918	. . 3 ⊢ ((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅}))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅})
4		uniun 4911	. . 3 ⊢ ∪ (ran 𝐹 ∪ {∅}) = (∪ ran 𝐹 ∪ ∪ {∅})
5		0ex 5282	. . . . 5 ⊢ ∅ ∈ V
6	5	unisn 4907	. . . 4 ⊢ ∪ {∅} = ∅
7	6	uneq2i 4145	. . 3 ⊢ (∪ ran 𝐹 ∪ ∪ {∅}) = (∪ ran 𝐹 ∪ ∅)
8		un0 4374	. . 3 ⊢ (∪ ran 𝐹 ∪ ∅) = ∪ ran 𝐹
9	4, 7, 8	3eqtri 2763	. 2 ⊢ ∪ (ran 𝐹 ∪ {∅}) = ∪ ran 𝐹
10	3, 9	sseqtri 4012	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ∈ wcel 2109   ∪ cun 3929   ⊆ wss 3931  ∅c0 4313  {csn 4606  ∪ cuni 4888  ran crn 5660  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-cnv 5667  df-dm 5669  df-rn 5670  df-iota 6489  df-fv 6544

This theorem is referenced by: (None)