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Theorem fvssunirnOLD 6924
Description: Obsolete version of fvssunirn 6923 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 (𝐹𝑋) ⊆ ran 𝐹

Proof of Theorem fvssunirnOLD
StepHypRef Expression
1 fvrn0 6920 . . 3 (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
2 elssuni 4936 . . 3 ((𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹𝑋) ⊆ (ran 𝐹 ∪ {∅}))
31, 2ax-mp 5 . 2 (𝐹𝑋) ⊆ (ran 𝐹 ∪ {∅})
4 uniun 4929 . . 3 (ran 𝐹 ∪ {∅}) = ( ran 𝐹 {∅})
5 0ex 5303 . . . . 5 ∅ ∈ V
65unisn 4925 . . . 4 {∅} = ∅
76uneq2i 4154 . . 3 ( ran 𝐹 {∅}) = ( ran 𝐹 ∪ ∅)
8 un0 4387 . . 3 ( ran 𝐹 ∪ ∅) = ran 𝐹
94, 7, 83eqtri 2757 . 2 (ran 𝐹 ∪ {∅}) = ran 𝐹
103, 9sseqtri 4010 1 (𝐹𝑋) ⊆ ran 𝐹
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  cun 3939  wss 3941  c0 4319  {csn 4625   cuni 4904  ran crn 5674  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-cnv 5681  df-dm 5683  df-rn 5684  df-iota 6495  df-fv 6551
This theorem is referenced by: (None)
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