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Mirrors > Home > MPE Home > Th. List > Mathboxes > mof02 | Structured version Visualization version GIF version |
Description: A variant of mof0 47592. (Contributed by Zhi Wang, 20-Sep-2024.) |
Ref | Expression |
---|---|
mof02 | ⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mof0 47592 | . 2 ⊢ ∃*𝑓 𝑓:𝐴⟶∅ | |
2 | feq3 6700 | . . 3 ⊢ (𝐵 = ∅ → (𝑓:𝐴⟶𝐵 ↔ 𝑓:𝐴⟶∅)) | |
3 | 2 | mobidv 2542 | . 2 ⊢ (𝐵 = ∅ → (∃*𝑓 𝑓:𝐴⟶𝐵 ↔ ∃*𝑓 𝑓:𝐴⟶∅)) |
4 | 1, 3 | mpbiri 258 | 1 ⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∃*wmo 2531 ∅c0 4322 ⟶wf 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-fun 6545 df-fn 6546 df-f 6547 |
This theorem is referenced by: mofsn2 47599 mofsssn 47600 mofmo 47601 |
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