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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem1 | Structured version Visualization version GIF version |
Description: Lemma for onsetrec 48247. (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 4915 | . . . 4 ⊢ (𝑥 = 𝑎 → ∪ 𝑥 = ∪ 𝑎) | |
2 | suceq 6431 | . . . . 5 ⊢ (∪ 𝑥 = ∪ 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑥 = 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎) |
4 | 1, 3 | preq12d 4742 | . . 3 ⊢ (𝑥 = 𝑎 → {∪ 𝑥, suc ∪ 𝑥} = {∪ 𝑎, suc ∪ 𝑎}) |
5 | onsetreclem1.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
6 | prex 5429 | . . 3 ⊢ {∪ 𝑎, suc ∪ 𝑎} ∈ V | |
7 | 4, 5, 6 | fvmpt 6998 | . 2 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}) |
8 | 7 | elv 3469 | 1 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 Vcvv 3463 {cpr 4627 ∪ cuni 4904 ↦ cmpt 5227 suc csuc 6367 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3944 df-un 3946 df-ss 3958 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-suc 6371 df-iota 6495 df-fun 6545 df-fv 6551 |
This theorem is referenced by: onsetreclem2 48245 onsetreclem3 48246 |
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