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Mirrors > Home > ILE Home > Th. List > relrnfvex | GIF version |
Description: If a function has a set range, then the function value exists unconditional on the domain. (Contributed by Mario Carneiro, 24-May-2019.) |
Ref | Expression |
---|---|
relrnfvex | ⊢ ((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relfvssunirn 5211 | . 2 ⊢ (Rel 𝐹 → (𝐹‘𝐴) ⊆ ∪ ran 𝐹) | |
2 | uniexg 4193 | . 2 ⊢ (ran 𝐹 ∈ V → ∪ ran 𝐹 ∈ V) | |
3 | ssexg 3917 | . 2 ⊢ (((𝐹‘𝐴) ⊆ ∪ ran 𝐹 ∧ ∪ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V) | |
4 | 1, 2, 3 | syl2an 283 | 1 ⊢ ((Rel 𝐹 ∧ ran 𝐹 ∈ V) → (𝐹‘𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1433 Vcvv 2601 ⊆ wss 2973 ∪ cuni 3601 ran crn 4364 Rel wrel 4368 ‘cfv 4922 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-dm 4373 df-rn 4374 df-iota 4887 df-fv 4930 |
This theorem is referenced by: (None) |
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