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Mirrors > Home > ILE Home > Th. List > tfrlemisucfn | GIF version |
Description: We can extend an acceptable function by one element to produce a function. Lemma for tfrlemi1 6229. (Contributed by Jim Kingdon, 2-Jul-2019.) |
Ref | Expression |
---|---|
tfrlemisucfn.1 | ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
tfrlemisucfn.2 | ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V)) |
tfrlemisucfn.3 | ⊢ (𝜑 → 𝑧 ∈ On) |
tfrlemisucfn.4 | ⊢ (𝜑 → 𝑔 Fn 𝑧) |
tfrlemisucfn.5 | ⊢ (𝜑 → 𝑔 ∈ 𝐴) |
Ref | Expression |
---|---|
tfrlemisucfn | ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) Fn suc 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2689 | . . 3 ⊢ 𝑧 ∈ V | |
2 | 1 | a1i 9 | . 2 ⊢ (𝜑 → 𝑧 ∈ V) |
3 | tfrlemisucfn.2 | . . . 4 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V)) | |
4 | 3 | tfrlem3-2d 6209 | . . 3 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)) |
5 | 4 | simprd 113 | . 2 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V) |
6 | tfrlemisucfn.4 | . 2 ⊢ (𝜑 → 𝑔 Fn 𝑧) | |
7 | eqid 2139 | . 2 ⊢ (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) | |
8 | df-suc 4293 | . 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧}) | |
9 | elirrv 4463 | . . 3 ⊢ ¬ 𝑧 ∈ 𝑧 | |
10 | 9 | a1i 9 | . 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧) |
11 | 2, 5, 6, 7, 8, 10 | fnunsn 5230 | 1 ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) Fn suc 𝑧) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∀wal 1329 = wceq 1331 ∈ wcel 1480 {cab 2125 ∀wral 2416 ∃wrex 2417 Vcvv 2686 ∪ cun 3069 {csn 3527 〈cop 3530 Oncon0 4285 suc csuc 4287 ↾ cres 4541 Fun wfun 5117 Fn wfn 5118 ‘cfv 5123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-suc 4293 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: tfrlemisucaccv 6222 tfrlemibfn 6225 |
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