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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 32397, 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 32397 | . 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 7394 | . . . . . . 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 6104 | . . . . 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 2622 | . . . 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 726 | . . 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 2726 | . 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 2635 | 1 ⊢ (𝑈↑↑𝑁) = {𝑦 ∣ (𝑁 ∈ ω ∧ ∅ = (rec(𝐹, 〈𝑁, 𝑦〉)‘𝑁))} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 Vcvv 3173 ∅c0 3874 ifcif 4036 〈cop 4131 ∪ cuni 4372 × cxp 5036 ‘cfv 5804 ↦ cmpt2 6551 ωcom 6957 1st c1st 7057 reccrdg 7392 1𝑜c1o 7440 ↑↑cfinxp 32396 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fv 5812 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-finxp 32397 |
This theorem is referenced by: finxpreclem6 32409 finxpsuclem 32410 |
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