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Theorem onsucrn 43261
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.)
Hypothesis
Ref Expression
onsucrn.f 𝐹 = (𝑥 ∈ On ↦ suc 𝑥)
Assertion
Ref Expression
onsucrn ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
Distinct variable group:   𝑎,𝑏,𝑥
Allowed substitution hints:   𝐹(𝑥,𝑎,𝑏)

Proof of Theorem onsucrn
StepHypRef Expression
1 simpr 484 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → 𝑎 = suc 𝑥)
2 onsuc 7831 . . . . . . . 8 (𝑥 ∈ On → suc 𝑥 ∈ On)
32adantr 480 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → suc 𝑥 ∈ On)
41, 3eqeltrd 2839 . . . . . 6 ((𝑥 ∈ On ∧ 𝑎 = suc 𝑥) → 𝑎 ∈ On)
54rexlimiva 3145 . . . . 5 (∃𝑥 ∈ On 𝑎 = suc 𝑥𝑎 ∈ On)
65pm4.71ri 560 . . . 4 (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ (𝑎 ∈ On ∧ ∃𝑥 ∈ On 𝑎 = suc 𝑥))
7 suceq 6452 . . . . . . 7 (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
87eqeq2d 2746 . . . . . 6 (𝑥 = 𝑏 → (𝑎 = suc 𝑥𝑎 = suc 𝑏))
98cbvrexvw 3236 . . . . 5 (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ ∃𝑏 ∈ On 𝑎 = suc 𝑏)
109anbi2i 623 . . . 4 ((𝑎 ∈ On ∧ ∃𝑥 ∈ On 𝑎 = suc 𝑥) ↔ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏))
116, 10bitri 275 . . 3 (∃𝑥 ∈ On 𝑎 = suc 𝑥 ↔ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏))
1211abbii 2807 . 2 {𝑎 ∣ ∃𝑥 ∈ On 𝑎 = suc 𝑥} = {𝑎 ∣ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏)}
13 onsucrn.f . . 3 𝐹 = (𝑥 ∈ On ↦ suc 𝑥)
1413rnmpt 5971 . 2 ran 𝐹 = {𝑎 ∣ ∃𝑥 ∈ On 𝑎 = suc 𝑥}
15 df-rab 3434 . 2 {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏} = {𝑎 ∣ (𝑎 ∈ On ∧ ∃𝑏 ∈ On 𝑎 = suc 𝑏)}
1612, 14, 153eqtr4i 2773 1 ran 𝐹 = {𝑎 ∈ On ∣ ∃𝑏 ∈ On 𝑎 = suc 𝑏}
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  {cab 2712  wrex 3068  {crab 3433  cmpt 5231  ran crn 5690  Oncon0 6386  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-cnv 5697  df-dm 5699  df-rn 5700  df-ord 6389  df-on 6390  df-suc 6392
This theorem is referenced by:  onsucf1o  43262
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