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Theorem fvssunirnOLD 6876
Description: Obsolete version of fvssunirn 6875 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 6872 . . 3 (𝐹𝑋) ∈ (ran 𝐹 ∪ {∅})
2 elssuni 4898 . . 3 ((𝐹𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹𝑋) ⊆ (ran 𝐹 ∪ {∅}))
31, 2ax-mp 5 . 2 (𝐹𝑋) ⊆ (ran 𝐹 ∪ {∅})
4 uniun 4891 . . 3 (ran 𝐹 ∪ {∅}) = ( ran 𝐹 {∅})
5 0ex 5264 . . . . 5 ∅ ∈ V
65unisn 4887 . . . 4 {∅} = ∅
76uneq2i 4120 . . 3 ( ran 𝐹 {∅}) = ( ran 𝐹 ∪ ∅)
8 un0 4350 . . 3 ( ran 𝐹 ∪ ∅) = ran 𝐹
94, 7, 83eqtri 2768 . 2 (ran 𝐹 ∪ {∅}) = ran 𝐹
103, 9sseqtri 3980 1 (𝐹𝑋) ⊆ ran 𝐹
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  cun 3908  wss 3910  c0 4282  {csn 4586   cuni 4865  ran crn 5634  cfv 6496
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-cnv 5641  df-dm 5643  df-rn 5644  df-iota 6448  df-fv 6504
This theorem is referenced by: (None)
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