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Mirrors > Home > ILE Home > Th. List > rdgfun | GIF version |
Description: The recursive definition generator is a function. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
rdgfun | ⊢ Fun rec(𝐹, 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . 3 ⊢ {𝑓 ∣ ∃𝑦 ∈ On (𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘(𝑓 ↾ 𝑧)))} = {𝑓 ∣ ∃𝑦 ∈ On (𝑓 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑓‘𝑧) = ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘(𝑓 ↾ 𝑧)))} | |
2 | 1 | tfrlem7 6317 | . 2 ⊢ Fun recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))) |
3 | df-irdg 6370 | . . 3 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))) | |
4 | 3 | funeqi 5237 | . 2 ⊢ (Fun rec(𝐹, 𝐴) ↔ Fun recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))) |
5 | 2, 4 | mpbir 146 | 1 ⊢ Fun rec(𝐹, 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 {cab 2163 ∀wral 2455 ∃wrex 2456 Vcvv 2737 ∪ cun 3127 ∪ ciun 3886 ↦ cmpt 4064 Oncon0 4363 dom cdm 4626 ↾ cres 4628 Fun wfun 5210 Fn wfn 5211 ‘cfv 5216 recscrecs 6304 reccrdg 6369 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-recs 6305 df-irdg 6370 |
This theorem is referenced by: rdgivallem 6381 |
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