Theorem fvssunirnOLD 6925

Description: Obsolete version of fvssunirn 6924 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 6921	. . 3 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅})
2		elssuni 4935	. . 3 ⊢ ((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅}))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅})
4		uniun 4928	. . 3 ⊢ ∪ (ran 𝐹 ∪ {∅}) = (∪ ran 𝐹 ∪ ∪ {∅})
5		0ex 5301	. . . . 5 ⊢ ∅ ∈ V
6	5	unisn 4924	. . . 4 ⊢ ∪ {∅} = ∅
7	6	uneq2i 4156	. . 3 ⊢ (∪ ran 𝐹 ∪ ∪ {∅}) = (∪ ran 𝐹 ∪ ∅)
8		un0 4386	. . 3 ⊢ (∪ ran 𝐹 ∪ ∅) = ∪ ran 𝐹
9	4, 7, 8	3eqtri 2759	. 2 ⊢ ∪ (ran 𝐹 ∪ {∅}) = ∪ ran 𝐹
10	3, 9	sseqtri 4014	1 ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹

Syntax hints:   ∈ wcel 2099   ∪ cun 3942   ⊆ wss 3944  ∅c0 4318  {csn 4624  ∪ cuni 4903  ran crn 5673  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-cnv 5680  df-dm 5682  df-rn 5683  df-iota 6494  df-fv 6550

This theorem is referenced by: (None)