![]() |
Mathbox for ML |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dffinxpf | Structured version Visualization version GIF version |
Description: This theorem is the same as the definition df-finxp 34217, except that the large function is replaced by a class variable for brevity. (Contributed by ML, 24-Oct-2020.) |
Ref | Expression |
---|---|
dffinxpf.1 | ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) |
Ref | Expression |
---|---|
dffinxpf | ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁))} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-finxp 34217 | . 2 ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈𝑁, 𝑦〉)‘𝑁))} | |
2 | dffinxpf.1 | . . . . . . 7 ⊢ 𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) | |
3 | rdgeq1 7906 | . . . . . . 7 ⊢ (𝐹 = (𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))) → rec(𝐹, 〈𝑁, 𝑦〉) = rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈𝑁, 𝑦〉)) | |
4 | 2, 3 | ax-mp 5 | . . . . . 6 ⊢ rec(𝐹, 〈𝑁, 𝑦〉) = rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈𝑁, 𝑦〉) |
5 | 4 | fveq1i 6546 | . . . . 5 ⊢ (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁) = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈𝑁, 𝑦〉)‘𝑁) |
6 | 5 | eqeq2i 2809 | . . . 4 ⊢ (∅ = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁) ↔ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈𝑁, 𝑦〉)‘𝑁)) |
7 | 6 | anbi2i 622 | . . 3 ⊢ ((𝑁 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁)) ↔ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈𝑁, 𝑦〉)‘𝑁))) |
8 | 7 | abbii 2863 | . 2 ⊢ {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁))} = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec((𝑛 ∈ ω, 𝑥 ∈ V ↦ if((𝑛 = 1o ∧ 𝑥 ∈ 𝑈), ∅, if(𝑥 ∈ (V × 𝑈), 〈∪ 𝑛, (1st ‘𝑥)〉, 〈𝑛, 𝑥〉))), 〈𝑁, 𝑦〉)‘𝑁))} |
9 | 1, 8 | eqtr4i 2824 | 1 ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁))} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1525 ∈ wcel 2083 {cab 2777 Vcvv 3440 ∅c0 4217 ifcif 4387 〈cop 4484 ∪ cuni 4751 × cxp 5448 ‘cfv 6232 ∈ cmpo 7025 ωcom 7443 1st c1st 7550 reccrdg 7904 1oc1o 7953 ↑↑cfinxp 34216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-ext 2771 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-br 4969 df-opab 5031 df-mpt 5048 df-xp 5456 df-cnv 5458 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-iota 6196 df-fv 6240 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-finxp 34217 |
This theorem is referenced by: finxpreclem6 34229 finxpsuclem 34230 |
Copyright terms: Public domain | W3C validator |