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| Mirrors > Home > MPE Home > Th. List > Mathboxes > onsucrn | Structured version Visualization version GIF version | ||
| Description: The successor operation is surjective onto its range, the class of successor ordinals. Lemma 1.17 of [Schloeder] p. 2. (Contributed by RP, 18-Jan-2025.) |
| Ref | Expression |
|---|---|
| onsucrn.f | ⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥) |
| Ref | Expression |
|---|---|
| onsucrn | ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpr 484 | . . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → 𝑎 = suc 𝑥) | |
| 2 | onsuc 7831 | . . . . . . . 8 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On) | |
| 3 | 2 | adantr 480 | . . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → suc 𝑥 ∈ On) |
| 4 | 1, 3 | eqeltrd 2841 | . . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → 𝑎 ∈ On) |
| 5 | 4 | rexlimiva 3147 | . . . . 5 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 → 𝑎 ∈ On) |
| 6 | 5 | pm4.71ri 560 | . . . 4 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ (𝑎 ∈ On ∧ ∃𝑥 ∈ On 𝑎 = suc 𝑥)) |
| 7 | suceq 6450 | . . . . . . 7 ⊢ (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏) | |
| 8 | 7 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑥 = 𝑏 → (𝑎 = suc 𝑥 ↔ 𝑎 = suc 𝑏)) |
| 9 | 8 | cbvrexvw 3238 | . . . . 5 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ ∃𝑏 ∈ On 𝑎 = suc 𝑏) |
| 10 | 9 | anbi2i 623 | . . . 4 ⊢ ((𝑎 ∈ On ∧ ∃𝑥 ∈ On 𝑎 = suc 𝑥) ↔ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏)) |
| 11 | 6, 10 | bitri 275 | . . 3 ⊢ (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏)) |
| 12 | 11 | abbii 2809 | . 2 ⊢ {𝑎 ∣ ∃𝑥 ∈ On 𝑎 = suc 𝑥} = {𝑎 ∣ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏)} |
| 13 | onsucrn.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ On ↦ suc 𝑥) | |
| 14 | 13 | rnmpt 5968 | . 2 ⊢ ran 𝐹 = {𝑎 ∣ ∃𝑥 ∈ On 𝑎 = suc 𝑥} |
| 15 | df-rab 3437 | . 2 ⊢ {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} = {𝑎 ∣ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏)} | |
| 16 | 12, 14, 15 | 3eqtr4i 2775 | 1 ⊢ ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 {crab 3436 ↦ cmpt 5225 ran crn 5686 Oncon0 6384 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-cnv 5693 df-dm 5695 df-rn 5696 df-ord 6387 df-on 6388 df-suc 6390 |
| This theorem is referenced by: onsucf1o 43285 |
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