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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 6237. (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 2692 | . . 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 6217 | . . 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 2140 | . 2 ⊢ (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) | |
8 | df-suc 4301 | . 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧}) | |
9 | elirrv 4471 | . . 3 ⊢ ¬ 𝑧 ∈ 𝑧 | |
10 | 9 | a1i 9 | . 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧) |
11 | 2, 5, 6, 7, 8, 10 | fnunsn 5238 | 1 ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) Fn suc 𝑧) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∀wal 1330 = wceq 1332 ∈ wcel 1481 {cab 2126 ∀wral 2417 ∃wrex 2418 Vcvv 2689 ∪ cun 3074 {csn 3532 〈cop 3535 Oncon0 4293 suc csuc 4295 ↾ cres 4549 Fun wfun 5125 Fn wfn 5126 ‘cfv 5131 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fn 5134 df-fv 5139 |
This theorem is referenced by: tfrlemisucaccv 6230 tfrlemibfn 6233 |
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