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Mirrors > Home > MPE Home > Th. List > Mathboxes > onsetreclem1 | Structured version Visualization version GIF version |
Description: Lemma for onsetrec 47239. (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 4877 | . . . 4 ⊢ (𝑥 = 𝑎 → ∪ 𝑥 = ∪ 𝑎) | |
2 | suceq 6384 | . . . . 5 ⊢ (∪ 𝑥 = ∪ 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝑥 = 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎) |
4 | 1, 3 | preq12d 4703 | . . 3 ⊢ (𝑥 = 𝑎 → {∪ 𝑥, suc ∪ 𝑥} = {∪ 𝑎, suc ∪ 𝑎}) |
5 | onsetreclem1.1 | . . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥}) | |
6 | prex 5390 | . . 3 ⊢ {∪ 𝑎, suc ∪ 𝑎} ∈ V | |
7 | 4, 5, 6 | fvmpt 6949 | . 2 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}) |
8 | 7 | elv 3450 | 1 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3444 {cpr 4589 ∪ cuni 4866 ↦ cmpt 5189 suc csuc 6320 ‘cfv 6497 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-suc 6324 df-iota 6449 df-fun 6499 df-fv 6505 |
This theorem is referenced by: onsetreclem2 47237 onsetreclem3 47238 |
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