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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsucf1o | Structured version Visualization version GIF version |
Description: The successor operation is a bijective function between the ordinals and the class of succesor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.) |
Ref | Expression |
---|---|
onsucf1o.f | ⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥) |
Ref | Expression |
---|---|
onsucf1o | ⊢ 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsucf1o.f | . . . 4 ⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥) | |
2 | 1 | fin1a2lem2 10380 | . . 3 ⊢ 𝐹:On–1-1→On |
3 | f1fn 6776 | . . 3 ⊢ (𝐹:On–1-1→On → 𝐹 Fn On) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ 𝐹 Fn On |
5 | 1 | onsucrn 41856 | . 2 ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} |
6 | 1 | fin1a2lem1 10379 | . . . . . 6 ⊢ (𝑎 ∈ On → (𝐹‘𝑎) = suc 𝑎) |
7 | 1 | fin1a2lem1 10379 | . . . . . 6 ⊢ (𝑏 ∈ On → (𝐹‘𝑏) = suc 𝑏) |
8 | 6, 7 | eqeqan12d 2746 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ suc 𝑎 = suc 𝑏)) |
9 | suc11 6461 | . . . . 5 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → (suc 𝑎 = suc 𝑏 ↔ 𝑎 = 𝑏)) | |
10 | 8, 9 | bitrd 278 | . . . 4 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ 𝑎 = 𝑏)) |
11 | 10 | biimpd 228 | . . 3 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ On) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)) |
12 | 11 | rgen2 3197 | . 2 ⊢ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏) |
13 | dff1o6 7258 | . 2 ⊢ (𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ↔ (𝐹 Fn On ∧ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} ∧ ∀𝑎 ∈ On ∀𝑏 ∈ On ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))) | |
14 | 4, 5, 12, 13 | mpbir3an 1341 | 1 ⊢ 𝐹:On–1-1-onto→{𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 {crab 3432 ↦ cmpt 5225 ran crn 5671 Oncon0 6354 suc csuc 6356 Fn wfn 6528 –1-1→wf1 6530 –1-1-onto→wf1o 6532 ‘cfv 6533 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-ord 6357 df-on 6358 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 |
This theorem is referenced by: (None) |
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