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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 34667, 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 34667 | . 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 8049 | . . . . . . 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 6673 | . . . . 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 2836 | . . . 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 624 | . . 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 2888 | . 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 2849 | 1 ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁))} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 {cab 2801 Vcvv 3496 ∅c0 4293 ifcif 4469 〈cop 4575 ∪ cuni 4840 × cxp 5555 ‘cfv 6357 ∈ cmpo 7160 ωcom 7582 1st c1st 7689 reccrdg 8047 1oc1o 8097 ↑↑cfinxp 34666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-iota 6316 df-fv 6365 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-finxp 34667 |
This theorem is referenced by: finxpreclem6 34679 finxpsuclem 34680 |
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