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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fsetsnf1o | Structured version Visualization version GIF version |
Description: The mapping of an element of a class to a singleton function is a bijection. (Contributed by AV, 13-Sep-2024.) |
Ref | Expression |
---|---|
fsetsnf.a | ⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} |
fsetsnf.f | ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}) |
Ref | Expression |
---|---|
fsetsnf1o | ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1-onto→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsetsnf.a | . . 3 ⊢ 𝐴 = {𝑦 ∣ ∃𝑏 ∈ 𝐵 𝑦 = {⟨𝑆, 𝑏⟩}} | |
2 | fsetsnf.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ {⟨𝑆, 𝑥⟩}) | |
3 | 1, 2 | fsetsnf1 46497 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1→𝐴) |
4 | 1, 2 | fsetsnfo 46498 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–onto→𝐴) |
5 | df-f1o 6550 | . 2 ⊢ (𝐹:𝐵–1-1-onto→𝐴 ↔ (𝐹:𝐵–1-1→𝐴 ∧ 𝐹:𝐵–onto→𝐴)) | |
6 | 3, 4, 5 | sylanbrc 581 | 1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {cab 2702 ∃wrex 3060 {csn 4624 ⟨cop 4630 ↦ cmpt 5226 –1-1→wf1 6540 –onto→wfo 6541 –1-1-onto→wf1o 6542 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 |
This theorem is referenced by: fsetsnprcnex 46500 |
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