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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem1 | Structured version Visualization version GIF version |
Description: Lemma for onsetrec 48939. (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 4923 | . . . 4 ⊢ (𝑥 = 𝑎 → ∪ 𝑥 = ∪ 𝑎) | |
2 | suceq 6452 | . . . . 5 ⊢ (∪ 𝑥 = ∪ 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑥 = 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎) |
4 | 1, 3 | preq12d 4746 | . . 3 ⊢ (𝑥 = 𝑎 → {∪ 𝑥, suc ∪ 𝑥} = {∪ 𝑎, suc ∪ 𝑎}) |
5 | onsetreclem1.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
6 | prex 5443 | . . 3 ⊢ {∪ 𝑎, suc ∪ 𝑎} ∈ V | |
7 | 4, 5, 6 | fvmpt 7016 | . 2 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}) |
8 | 7 | elv 3483 | 1 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 Vcvv 3478 {cpr 4633 ∪ cuni 4912 ↦ cmpt 5231 suc csuc 6388 ‘cfv 6563 |
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 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 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-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-suc 6392 df-iota 6516 df-fun 6565 df-fv 6571 |
This theorem is referenced by: onsetreclem2 48937 onsetreclem3 48938 |
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