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Mirrors > Home > MPE Home > Th. List > fvfundmfvn0 | Structured version Visualization version GIF version |
Description: If the "value of a class" at an argument is not the empty set, then the argument is in the domain of the class and the class restricted to the singleton formed on that argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.) (Proof shortened by BJ, 13-Aug-2022.) |
Ref | Expression |
---|---|
fvfundmfvn0 | ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndmfv 6920 | . . 3 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅) | |
2 | 1 | necon1ai 2962 | . 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → 𝐴 ∈ dom 𝐹) |
3 | nfunsn 6927 | . . 3 ⊢ (¬ Fun (𝐹 ↾ {𝐴}) → (𝐹‘𝐴) = ∅) | |
4 | 3 | necon1ai 2962 | . 2 ⊢ ((𝐹‘𝐴) ≠ ∅ → Fun (𝐹 ↾ {𝐴})) |
5 | 2, 4 | jca 511 | 1 ⊢ ((𝐹‘𝐴) ≠ ∅ → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 ≠ wne 2934 ∅c0 4317 {csn 4623 dom cdm 5669 ↾ cres 5671 Fun wfun 6531 ‘cfv 6537 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-res 5681 df-iota 6489 df-fun 6539 df-fv 6545 |
This theorem is referenced by: fvn0ssdmfun 7070 fvn0fvelrnOLD 7157 umgrnloopv 28874 usgrnloopvALT 28966 afvpcfv0 46431 afvfvn0fveq 46435 afv0nbfvbi 46436 afv2fvn0fveq 46549 ovn0dmfun 47111 |
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