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Mirrors > Home > ILE Home > Th. List > tfrlemibex | GIF version |
Description: The set 𝐵 exists. Lemma for tfrlemi1 6311. (Contributed by Jim Kingdon, 17-Mar-2019.) (Proof shortened by Mario Carneiro, 24-May-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)) |
tfrlemi1.3 | ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))} |
tfrlemi1.4 | ⊢ (𝜑 → 𝑥 ∈ On) |
tfrlemi1.5 | ⊢ (𝜑 → ∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) |
Ref | Expression |
---|---|
tfrlemibex | ⊢ (𝜑 → 𝐵 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tfrlemisucfn.1 | . . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} | |
2 | tfrlemisucfn.2 | . . . 4 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V)) | |
3 | tfrlemi1.3 | . . . 4 ⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))} | |
4 | tfrlemi1.4 | . . . 4 ⊢ (𝜑 → 𝑥 ∈ On) | |
5 | tfrlemi1.5 | . . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) | |
6 | 1, 2, 3, 4, 5 | tfrlemibfn 6307 | . . 3 ⊢ (𝜑 → ∪ 𝐵 Fn 𝑥) |
7 | vex 2733 | . . 3 ⊢ 𝑥 ∈ V | |
8 | fnex 5718 | . . 3 ⊢ ((∪ 𝐵 Fn 𝑥 ∧ 𝑥 ∈ V) → ∪ 𝐵 ∈ V) | |
9 | 6, 7, 8 | sylancl 411 | . 2 ⊢ (𝜑 → ∪ 𝐵 ∈ V) |
10 | uniexb 4458 | . 2 ⊢ (𝐵 ∈ V ↔ ∪ 𝐵 ∈ V) | |
11 | 9, 10 | sylibr 133 | 1 ⊢ (𝜑 → 𝐵 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 973 ∀wal 1346 = wceq 1348 ∃wex 1485 ∈ wcel 2141 {cab 2156 ∀wral 2448 ∃wrex 2449 Vcvv 2730 ∪ cun 3119 {csn 3583 〈cop 3586 ∪ cuni 3796 Oncon0 4348 ↾ 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-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 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-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 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-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-recs 6284 |
This theorem is referenced by: tfrlemiex 6310 |
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