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Mirrors > Home > MPE Home > Th. List > Mathboxes > mof02 | Structured version Visualization version GIF version |
Description: A variant of mof0 45742. (Contributed by Zhi Wang, 20-Sep-2024.) |
Ref | Expression |
---|---|
mof02 | ⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mof0 45742 | . 2 ⊢ ∃*𝑓 𝑓:𝐴⟶∅ | |
2 | feq3 6497 | . . 3 ⊢ (𝐵 = ∅ → (𝑓:𝐴⟶𝐵 ↔ 𝑓:𝐴⟶∅)) | |
3 | 2 | mobidv 2550 | . 2 ⊢ (𝐵 = ∅ → (∃*𝑓 𝑓:𝐴⟶𝐵 ↔ ∃*𝑓 𝑓:𝐴⟶∅)) |
4 | 1, 3 | mpbiri 261 | 1 ⊢ (𝐵 = ∅ → ∃*𝑓 𝑓:𝐴⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∃*wmo 2539 ∅c0 4221 ⟶wf 6345 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-rex 3060 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-fun 6351 df-fn 6352 df-f 6353 |
This theorem is referenced by: mofsn2 45749 mofsssn 45750 mofmo 45751 |
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