Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 6311. (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 2733 | . . 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 6291 | . . 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 2170 | . 2 ⊢ (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) | |
8 | df-suc 4356 | . 2 ⊢ suc 𝑧 = (𝑧 ∪ {𝑧}) | |
9 | elirrv 4532 | . . 3 ⊢ ¬ 𝑧 ∈ 𝑧 | |
10 | 9 | a1i 9 | . 2 ⊢ (𝜑 → ¬ 𝑧 ∈ 𝑧) |
11 | 2, 5, 6, 7, 8, 10 | fnunsn 5305 | 1 ⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) Fn suc 𝑧) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∀wal 1346 = wceq 1348 ∈ wcel 2141 {cab 2156 ∀wral 2448 ∃wrex 2449 Vcvv 2730 ∪ cun 3119 {csn 3583 〈cop 3586 Oncon0 4348 suc csuc 4350 ↾ cres 4613 Fun wfun 5192 Fn wfn 5193 ‘cfv 5198 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-id 4278 df-suc 4356 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-iota 5160 df-fun 5200 df-fn 5201 df-fv 5206 |
This theorem is referenced by: tfrlemisucaccv 6304 tfrlemibfn 6307 |
Copyright terms: Public domain | W3C validator |