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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 46334 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1→𝐴) |
4 | 1, 2 | fsetsnfo 46335 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–onto→𝐴) |
5 | df-f1o 6544 | . 2 ⊢ (𝐹:𝐵–1-1-onto→𝐴 ↔ (𝐹:𝐵–1-1→𝐴 ∧ 𝐹:𝐵–onto→𝐴)) | |
6 | 3, 4, 5 | sylanbrc 582 | 1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 {cab 2703 ∃wrex 3064 {csn 4623 ⟨cop 4629 ↦ cmpt 5224 –1-1→wf1 6534 –onto→wfo 6535 –1-1-onto→wf1o 6536 |
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-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 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-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 |
This theorem is referenced by: fsetsnprcnex 46337 |
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