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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem1 | Structured version Visualization version GIF version |
Description: Lemma for onsetrec 46413. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
onsetreclem1.1 | ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) |
Ref | Expression |
---|---|
onsetreclem1 | ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unieq 4850 | . . . 4 ⊢ (𝑥 = 𝑎 → ∪ 𝑥 = ∪ 𝑎) | |
2 | suceq 6331 | . . . . 5 ⊢ (∪ 𝑥 = ∪ 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑥 = 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎) |
4 | 1, 3 | preq12d 4677 | . . 3 ⊢ (𝑥 = 𝑎 → {∪ 𝑥, suc ∪ 𝑥} = {∪ 𝑎, suc ∪ 𝑎}) |
5 | onsetreclem1.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
6 | prex 5355 | . . 3 ⊢ {∪ 𝑎, suc ∪ 𝑎} ∈ V | |
7 | 4, 5, 6 | fvmpt 6875 | . 2 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}) |
8 | 7 | elv 3438 | 1 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 Vcvv 3432 {cpr 4563 ∪ cuni 4839 ↦ cmpt 5157 suc csuc 6268 ‘cfv 6433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-suc 6272 df-iota 6391 df-fun 6435 df-fv 6441 |
This theorem is referenced by: onsetreclem2 46411 onsetreclem3 46412 |
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