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Mirrors > Home > ILE Home > Th. List > tfrlem3a | GIF version |
Description: Lemma for transfinite recursion. Let A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in A for later use. (Contributed by NM, 9-Apr-1995.) |
Ref | Expression |
---|---|
tfrlem3.1 | ⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))} |
tfrlem3.2 | ⊢ 𝐺 ∈ V |
Ref | Expression |
---|---|
tfrlem3a | ⊢ (𝐺 ∈ A ↔ ∃z ∈ On (𝐺 Fn z ∧ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlem3.2 | . 2 ⊢ 𝐺 ∈ V | |
2 | fneq12 4935 | . . . 4 ⊢ ((f = 𝐺 ∧ x = z) → (f Fn x ↔ 𝐺 Fn z)) | |
3 | simpll 481 | . . . . . . 7 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → f = 𝐺) | |
4 | simpr 103 | . . . . . . 7 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → y = w) | |
5 | 3, 4 | fveq12d 5127 | . . . . . 6 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → (f‘y) = (𝐺‘w)) |
6 | 3, 4 | reseq12d 4556 | . . . . . . 7 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → (f ↾ y) = (𝐺 ↾ w)) |
7 | 6 | fveq2d 5125 | . . . . . 6 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → (𝐹‘(f ↾ y)) = (𝐹‘(𝐺 ↾ w))) |
8 | 5, 7 | eqeq12d 2051 | . . . . 5 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → ((f‘y) = (𝐹‘(f ↾ y)) ↔ (𝐺‘w) = (𝐹‘(𝐺 ↾ w)))) |
9 | simpr 103 | . . . . . 6 ⊢ ((f = 𝐺 ∧ x = z) → x = z) | |
10 | 9 | adantr 261 | . . . . 5 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → x = z) |
11 | 8, 10 | cbvraldva2 2531 | . . . 4 ⊢ ((f = 𝐺 ∧ x = z) → (∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)) ↔ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w)))) |
12 | 2, 11 | anbi12d 442 | . . 3 ⊢ ((f = 𝐺 ∧ x = z) → ((f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y))) ↔ (𝐺 Fn z ∧ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w))))) |
13 | 12 | cbvrexdva 2534 | . 2 ⊢ (f = 𝐺 → (∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y))) ↔ ∃z ∈ On (𝐺 Fn z ∧ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w))))) |
14 | tfrlem3.1 | . 2 ⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))} | |
15 | 1, 13, 14 | elab2 2684 | 1 ⊢ (𝐺 ∈ A ↔ ∃z ∈ On (𝐺 Fn z ∧ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w)))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ wcel 1390 {cab 2023 ∀wral 2300 ∃wrex 2301 Vcvv 2551 Oncon0 4066 ↾ cres 4290 Fn wfn 4840 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-res 4300 df-iota 4810 df-fun 4847 df-fn 4848 df-fv 4853 |
This theorem is referenced by: tfrlem3 5867 tfrlem5 5871 tfrlemisucaccv 5880 tfrlemibxssdm 5882 tfrlemi14d 5888 tfrexlem 5889 |
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