![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > frrlem10 | Structured version Visualization version GIF version |
Description: Lemma for founded recursion. The union of all acceptable functions is a function. (Contributed by Paul Chapman, 21-Apr-2012.) (Revised by Scott Fenton, 23-Dec-2021.) |
Ref | Expression |
---|---|
frrlem10.1 | ⊢ 𝑅 Fr 𝐴 |
frrlem10.2 | ⊢ 𝑅 Se 𝐴 |
frrlem10.3 | ⊢ 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))))} |
frrlem10.4 | ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
frrlem10 | ⊢ Fun 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3841 | . . 3 ⊢ 𝐵 ⊆ 𝐵 | |
2 | frrlem10.1 | . . . 4 ⊢ 𝑅 Fr 𝐴 | |
3 | frrlem10.2 | . . . 4 ⊢ 𝑅 Se 𝐴 | |
4 | frrlem10.3 | . . . 4 ⊢ 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))))} | |
5 | 2, 3, 4 | frrlem5c 32375 | . . 3 ⊢ (𝐵 ⊆ 𝐵 → Fun ∪ 𝐵) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ Fun ∪ 𝐵 |
7 | df-frecs 32365 | . . . 4 ⊢ frecs(𝑅, 𝐴, 𝐺) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} | |
8 | frrlem10.4 | . . . 4 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) | |
9 | 4 | unieqi 4680 | . . . 4 ⊢ ∪ 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))))} |
10 | 7, 8, 9 | 3eqtr4i 2811 | . . 3 ⊢ 𝐹 = ∪ 𝐵 |
11 | 10 | funeqi 6156 | . 2 ⊢ (Fun 𝐹 ↔ Fun ∪ 𝐵) |
12 | 6, 11 | mpbir 223 | 1 ⊢ Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∃wex 1823 {cab 2762 ∀wral 3089 ⊆ wss 3791 ∪ cuni 4671 Fr wfr 5311 Se wse 5312 ↾ cres 5357 Predcpred 5932 Fun wfun 6129 Fn wfn 6130 ‘cfv 6135 (class class class)co 6922 frecscfrecs 32364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-ov 6925 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-trpred 32306 df-frecs 32365 |
This theorem is referenced by: frrlem11 32381 |
Copyright terms: Public domain | W3C validator |