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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem2 | Structured version Visualization version GIF version |
Description: Lemma for onsetrec 42250. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
onsetreclem2.1 | ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) |
Ref | Expression |
---|---|
onsetreclem2 | ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onsetreclem2.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
2 | 1 | onsetreclem1 42247 | . 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
3 | vex 3176 | . . . 4 ⊢ 𝑎 ∈ V | |
4 | 3 | ssonunii 6879 | . . 3 ⊢ (𝑎 ⊆ On → ∪ 𝑎 ∈ On) |
5 | suceloni 6905 | . . . 4 ⊢ (∪ 𝑎 ∈ On → suc ∪ 𝑎 ∈ On) | |
6 | prssi 4293 | . . . 4 ⊢ ((∪ 𝑎 ∈ On ∧ suc ∪ 𝑎 ∈ On) → {∪ 𝑎, suc ∪ 𝑎} ⊆ On) | |
7 | 5, 6 | mpdan 699 | . . 3 ⊢ (∪ 𝑎 ∈ On → {∪ 𝑎, suc ∪ 𝑎} ⊆ On) |
8 | 4, 7 | syl 17 | . 2 ⊢ (𝑎 ⊆ On → {∪ 𝑎, suc ∪ 𝑎} ⊆ On) |
9 | 2, 8 | syl5eqss 3612 | 1 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 {cpr 4127 ∪ cuni 4372 ↦ cmpt 4643 Oncon0 5640 suc csuc 5642 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-ord 5643 df-on 5644 df-suc 5646 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: onsetrec 42250 |
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