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Mirrors > Home > ILE Home > Th. List > frectfr | GIF version |
Description: Lemma to connect
transfinite recursion theorems with finite recursion.
That is, given the conditions 𝐹 Fn V and 𝐴 ∈ 𝑉 on
frec(𝐹, 𝐴), we want to be able to apply tfri1d 6336 or tfri2d 6337,
and this lemma lets us satisfy hypotheses of those theorems.
(Contributed by Jim Kingdon, 15-Aug-2019.) |
Ref | Expression |
---|---|
frectfr.1 | ⊢ 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}) |
Ref | Expression |
---|---|
frectfr | ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺‘𝑦) ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2741 | . . . . . . . 8 ⊢ 𝑔 ∈ V | |
2 | 1 | a1i 9 | . . . . . . 7 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → 𝑔 ∈ V) |
3 | simpl 109 | . . . . . . 7 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → ∀𝑧(𝐹‘𝑧) ∈ V) | |
4 | simpr 110 | . . . . . . 7 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉) | |
5 | 2, 3, 4 | frecabex 6399 | . . . . . 6 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V) |
6 | 5 | ralrimivw 2551 | . . . . 5 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → ∀𝑔 ∈ V {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V) |
7 | frectfr.1 | . . . . . 6 ⊢ 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}) | |
8 | 7 | fnmpt 5343 | . . . . 5 ⊢ (∀𝑔 ∈ V {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ V → 𝐺 Fn V) |
9 | 6, 8 | syl 14 | . . . 4 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → 𝐺 Fn V) |
10 | vex 2741 | . . . 4 ⊢ 𝑦 ∈ V | |
11 | funfvex 5533 | . . . . 5 ⊢ ((Fun 𝐺 ∧ 𝑦 ∈ dom 𝐺) → (𝐺‘𝑦) ∈ V) | |
12 | 11 | funfni 5317 | . . . 4 ⊢ ((𝐺 Fn V ∧ 𝑦 ∈ V) → (𝐺‘𝑦) ∈ V) |
13 | 9, 10, 12 | sylancl 413 | . . 3 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → (𝐺‘𝑦) ∈ V) |
14 | 7 | funmpt2 5256 | . . 3 ⊢ Fun 𝐺 |
15 | 13, 14 | jctil 312 | . 2 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → (Fun 𝐺 ∧ (𝐺‘𝑦) ∈ V)) |
16 | 15 | alrimiv 1874 | 1 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺‘𝑦) ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∨ wo 708 ∀wal 1351 = wceq 1353 ∈ wcel 2148 {cab 2163 ∀wral 2455 ∃wrex 2456 Vcvv 2738 ∅c0 3423 ↦ cmpt 4065 suc csuc 4366 ωcom 4590 dom cdm 4627 Fun wfun 5211 Fn wfn 5212 ‘cfv 5217 |
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-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4119 ax-sep 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-iinf 4588 |
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-reu 2462 df-rab 2464 df-v 2740 df-sbc 2964 df-csb 3059 df-un 3134 df-in 3136 df-ss 3143 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-iun 3889 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-iom 4591 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-ima 4640 df-iota 5179 df-fun 5219 df-fn 5220 df-f 5221 df-f1 5222 df-fo 5223 df-f1o 5224 df-fv 5225 |
This theorem is referenced by: frecfnom 6402 |
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