Mathbox for Emmett Weisz |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem1 | Structured version Visualization version GIF version |
Description: Lemma for onsetrec 44738. (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 4838 | . . . 4 ⊢ (𝑥 = 𝑎 → ∪ 𝑥 = ∪ 𝑎) | |
2 | suceq 6249 | . . . . 5 ⊢ (∪ 𝑥 = ∪ 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑥 = 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎) |
4 | 1, 3 | preq12d 4669 | . . 3 ⊢ (𝑥 = 𝑎 → {∪ 𝑥, suc ∪ 𝑥} = {∪ 𝑎, suc ∪ 𝑎}) |
5 | onsetreclem1.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
6 | prex 5323 | . . 3 ⊢ {∪ 𝑎, suc ∪ 𝑎} ∈ V | |
7 | 4, 5, 6 | fvmpt 6761 | . 2 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}) |
8 | 7 | elv 3497 | 1 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 Vcvv 3492 {cpr 4559 ∪ cuni 4830 ↦ cmpt 5137 suc csuc 6186 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-suc 6190 df-iota 6307 df-fun 6350 df-fv 6356 |
This theorem is referenced by: onsetreclem2 44736 onsetreclem3 44737 |
Copyright terms: Public domain | W3C validator |