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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 47001 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1→𝐴) |
4 | 1, 2 | fsetsnfo 47002 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–onto→𝐴) |
5 | df-f1o 6569 | . 2 ⊢ (𝐹:𝐵–1-1-onto→𝐴 ↔ (𝐹:𝐵–1-1→𝐴 ∧ 𝐹:𝐵–onto→𝐴)) | |
6 | 3, 4, 5 | sylanbrc 583 | 1 ⊢ (𝑆 ∈ 𝑉 → 𝐹:𝐵–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 {cab 2711 ∃wrex 3067 {csn 4630 〈cop 4636 ↦ cmpt 5230 –1-1→wf1 6559 –onto→wfo 6560 –1-1-onto→wf1o 6561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 |
This theorem is referenced by: fsetsnprcnex 47004 |
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