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Mirrors > Home > MPE Home > Th. List > Mathboxes > cfsetsnfsetf1o | Structured version Visualization version GIF version |
Description: The mapping of the class of singleton functions into the class of constant functions is a bijection. (Contributed by AV, 14-Sep-2024.) |
Ref | Expression |
---|---|
cfsetsnfsetfv.f | ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} |
cfsetsnfsetfv.g | ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵} |
cfsetsnfsetfv.h | ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))) |
Ref | Expression |
---|---|
cfsetsnfsetf1o | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1-onto→𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cfsetsnfsetfv.f | . . 3 ⊢ 𝐹 = {𝑓 ∣ (𝑓:𝐴⟶𝐵 ∧ ∃𝑏 ∈ 𝐵 ∀𝑧 ∈ 𝐴 (𝑓‘𝑧) = 𝑏)} | |
2 | cfsetsnfsetfv.g | . . 3 ⊢ 𝐺 = {𝑥 ∣ 𝑥:{𝑌}⟶𝐵} | |
3 | cfsetsnfsetfv.h | . . 3 ⊢ 𝐻 = (𝑔 ∈ 𝐺 ↦ (𝑎 ∈ 𝐴 ↦ (𝑔‘𝑌))) | |
4 | 1, 2, 3 | cfsetsnfsetf1 44061 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1→𝐹) |
5 | 1, 2, 3 | cfsetsnfsetfo 44062 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–onto→𝐹) |
6 | df-f1o 6347 | . 2 ⊢ (𝐻:𝐺–1-1-onto→𝐹 ↔ (𝐻:𝐺–1-1→𝐹 ∧ 𝐻:𝐺–onto→𝐹)) | |
7 | 4, 5, 6 | sylanbrc 586 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑌 ∈ 𝐴) → 𝐻:𝐺–1-1-onto→𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 {cab 2735 ∀wral 3070 ∃wrex 3071 {csn 4525 ↦ cmpt 5116 ⟶wf 6336 –1-1→wf1 6337 –onto→wfo 6338 –1-1-onto→wf1o 6339 ‘cfv 6340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5160 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 |
This theorem is referenced by: fsetprcnexALT 44064 |
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