![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > frrlem7 | Structured version Visualization version GIF version |
Description: Lemma for well-founded recursion. The well-founded recursion generator's domain is a subclass of 𝐴. (Contributed by Scott Fenton, 27-Aug-2022.) |
Ref | Expression |
---|---|
frrlem5.1 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
frrlem5.2 | ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
frrlem7 | ⊢ dom 𝐹 ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frrlem5.1 | . . . . . . 7 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} | |
2 | frrlem5.2 | . . . . . . 7 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) | |
3 | 1, 2 | frrlem5 8305 | . . . . . 6 ⊢ 𝐹 = ∪ 𝐵 |
4 | 3 | dmeqi 5911 | . . . . 5 ⊢ dom 𝐹 = dom ∪ 𝐵 |
5 | dmuni 5921 | . . . . 5 ⊢ dom ∪ 𝐵 = ∪ 𝑔 ∈ 𝐵 dom 𝑔 | |
6 | 4, 5 | eqtri 2754 | . . . 4 ⊢ dom 𝐹 = ∪ 𝑔 ∈ 𝐵 dom 𝑔 |
7 | 6 | sseq1i 4008 | . . 3 ⊢ (dom 𝐹 ⊆ 𝐴 ↔ ∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴) |
8 | iunss 5053 | . . 3 ⊢ (∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴 ↔ ∀𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴) | |
9 | 7, 8 | bitri 274 | . 2 ⊢ (dom 𝐹 ⊆ 𝐴 ↔ ∀𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴) |
10 | 1 | frrlem3 8303 | . 2 ⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴) |
11 | 9, 10 | mprgbir 3058 | 1 ⊢ dom 𝐹 ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∃wex 1774 {cab 2703 ∀wral 3051 ⊆ wss 3947 ∪ cuni 4913 ∪ ciun 5001 dom cdm 5682 ↾ cres 5684 Predcpred 6311 Fn wfn 6549 ‘cfv 6554 (class class class)co 7424 frecscfrecs 8295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-iota 6506 df-fun 6556 df-fn 6557 df-fv 6562 df-ov 7427 df-frecs 8296 |
This theorem is referenced by: frrlem14 8314 frrdmss 8322 |
Copyright terms: Public domain | W3C validator |