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Mirrors > Home > MPE Home > Th. List > fvssunirnOLD | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fvssunirnOLD | ⊢ (𝐹‘𝑋) ⊆ ∪ ran 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvrn0 6872 | . . 3 ⊢ (𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) | |
2 | elssuni 4898 | . . 3 ⊢ ((𝐹‘𝑋) ∈ (ran 𝐹 ∪ {∅}) → (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅})) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐹‘𝑋) ⊆ ∪ (ran 𝐹 ∪ {∅}) |
4 | uniun 4891 | . . 3 ⊢ ∪ (ran 𝐹 ∪ {∅}) = (∪ ran 𝐹 ∪ ∪ {∅}) | |
5 | 0ex 5264 | . . . . 5 ⊢ ∅ ∈ V | |
6 | 5 | unisn 4887 | . . . 4 ⊢ ∪ {∅} = ∅ |
7 | 6 | uneq2i 4120 | . . 3 ⊢ (∪ ran 𝐹 ∪ ∪ {∅}) = (∪ ran 𝐹 ∪ ∅) |
8 | un0 4350 | . . 3 ⊢ (∪ ran 𝐹 ∪ ∅) = ∪ ran 𝐹 | |
9 | 4, 7, 8 | 3eqtri 2768 | . 2 ⊢ ∪ (ran 𝐹 ∪ {∅}) = ∪ ran 𝐹 |
10 | 3, 9 | sseqtri 3980 | 1 ⊢ (𝐹‘𝑋) ⊆ ∪ 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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