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Theorem relfvssunirn 5218
 Description: The result of a function value is always a subset of the union of the range, even if it is invalid and thus empty. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
Assertion
Ref Expression
relfvssunirn (Rel 𝐹 → (𝐹𝐴) ⊆ ran 𝐹)

Proof of Theorem relfvssunirn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 relelrn 4597 . . . . 5 ((Rel 𝐹𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹)
21ex 112 . . . 4 (Rel 𝐹 → (𝐴𝐹𝑥𝑥 ∈ ran 𝐹))
3 elssuni 3635 . . . 4 (𝑥 ∈ ran 𝐹𝑥 ran 𝐹)
42, 3syl6 33 . . 3 (Rel 𝐹 → (𝐴𝐹𝑥𝑥 ran 𝐹))
54alrimiv 1770 . 2 (Rel 𝐹 → ∀𝑥(𝐴𝐹𝑥𝑥 ran 𝐹))
6 fvss 5216 . 2 (∀𝑥(𝐴𝐹𝑥𝑥 ran 𝐹) → (𝐹𝐴) ⊆ ran 𝐹)
75, 6syl 14 1 (Rel 𝐹 → (𝐹𝐴) ⊆ ran 𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1257   ∈ wcel 1409   ⊆ wss 2944  ∪ cuni 3607   class class class wbr 3791  ran crn 4373  Rel wrel 4377  ‘cfv 4929 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-xp 4378  df-rel 4379  df-cnv 4380  df-dm 4382  df-rn 4383  df-iota 4894  df-fv 4937 This theorem is referenced by:  relrnfvex  5220
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