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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 6356. (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 2755 | . . 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 6336 | . . 3 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)) |
5 | 4 | simprd 114 | . 2 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V) |
6 | tfrlemisucfn.4 | . 2 ⊢ (𝜑 → 𝑔 Fn 𝑧) | |
7 | eqid 2189 | . 2 ⊢ (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) | |
8 | df-suc 4389 | . 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧}) | |
9 | elirrv 4565 | . . 3 ⊢ ¬ 𝑧 ∈ 𝑧 | |
10 | 9 | a1i 9 | . 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧) |
11 | 2, 5, 6, 7, 8, 10 | fnunsn 5342 | 1 ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) Fn suc 𝑧) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∀wal 1362 = wceq 1364 ∈ wcel 2160 {cab 2175 ∀wral 2468 ∃wrex 2469 Vcvv 2752 ∪ cun 3142 {csn 3607 〈cop 3610 Oncon0 4381 suc csuc 4383 ↾ cres 4646 Fun wfun 5229 Fn wfn 5230 ‘cfv 5235 |
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-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-suc 4389 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fn 5238 df-fv 5243 |
This theorem is referenced by: tfrlemisucaccv 6349 tfrlemibfn 6352 |
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