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Mirrors > Home > ILE Home > Th. List > enctlem | GIF version |
Description: Lemma for enct 12447. One direction of the biconditional. (Contributed by Jim Kingdon, 23-Dec-2023.) |
Ref | Expression |
---|---|
enctlem | ⊢ (𝐴 ≈ 𝐵 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1oex 6438 | . . . . 5 ⊢ 1o ∈ V | |
2 | 1 | enref 6778 | . . . 4 ⊢ 1o ≈ 1o |
3 | djuen 7223 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 1o ≈ 1o) → (𝐴 ⊔ 1o) ≈ (𝐵 ⊔ 1o)) | |
4 | 2, 3 | mpan2 425 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ⊔ 1o) ≈ (𝐵 ⊔ 1o)) |
5 | bren 6760 | . . 3 ⊢ ((𝐴 ⊔ 1o) ≈ (𝐵 ⊔ 1o) ↔ ∃ℎ ℎ:(𝐴 ⊔ 1o)–1-1-onto→(𝐵 ⊔ 1o)) | |
6 | 4, 5 | sylib 122 | . 2 ⊢ (𝐴 ≈ 𝐵 → ∃ℎ ℎ:(𝐴 ⊔ 1o)–1-1-onto→(𝐵 ⊔ 1o)) |
7 | f1ofo 5480 | . . . . . 6 ⊢ (ℎ:(𝐴 ⊔ 1o)–1-1-onto→(𝐵 ⊔ 1o) → ℎ:(𝐴 ⊔ 1o)–onto→(𝐵 ⊔ 1o)) | |
8 | 7 | ad2antlr 489 | . . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ ℎ:(𝐴 ⊔ 1o)–1-1-onto→(𝐵 ⊔ 1o)) ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → ℎ:(𝐴 ⊔ 1o)–onto→(𝐵 ⊔ 1o)) |
9 | foco 5460 | . . . . . 6 ⊢ ((ℎ:(𝐴 ⊔ 1o)–onto→(𝐵 ⊔ 1o) ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (ℎ ∘ 𝑓):ω–onto→(𝐵 ⊔ 1o)) | |
10 | vex 2752 | . . . . . . . 8 ⊢ ℎ ∈ V | |
11 | vex 2752 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
12 | 10, 11 | coex 5186 | . . . . . . 7 ⊢ (ℎ ∘ 𝑓) ∈ V |
13 | foeq1 5446 | . . . . . . 7 ⊢ (𝑔 = (ℎ ∘ 𝑓) → (𝑔:ω–onto→(𝐵 ⊔ 1o) ↔ (ℎ ∘ 𝑓):ω–onto→(𝐵 ⊔ 1o))) | |
14 | 12, 13 | spcev 2844 | . . . . . 6 ⊢ ((ℎ ∘ 𝑓):ω–onto→(𝐵 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)) |
15 | 9, 14 | syl 14 | . . . . 5 ⊢ ((ℎ:(𝐴 ⊔ 1o)–onto→(𝐵 ⊔ 1o) ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)) |
16 | 8, 15 | sylancom 420 | . . . 4 ⊢ (((𝐴 ≈ 𝐵 ∧ ℎ:(𝐴 ⊔ 1o)–1-1-onto→(𝐵 ⊔ 1o)) ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o)) |
17 | 16 | ex 115 | . . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ ℎ:(𝐴 ⊔ 1o)–1-1-onto→(𝐵 ⊔ 1o)) → (𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o))) |
18 | 17 | exlimdv 1829 | . 2 ⊢ ((𝐴 ≈ 𝐵 ∧ ℎ:(𝐴 ⊔ 1o)–1-1-onto→(𝐵 ⊔ 1o)) → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o))) |
19 | 6, 18 | exlimddv 1908 | 1 ⊢ (𝐴 ≈ 𝐵 → (∃𝑓 𝑓:ω–onto→(𝐴 ⊔ 1o) → ∃𝑔 𝑔:ω–onto→(𝐵 ⊔ 1o))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∃wex 1502 class class class wbr 4015 ωcom 4601 ∘ ccom 4642 –onto→wfo 5226 –1-1-onto→wf1o 5227 1oc1o 6423 ≈ cen 6751 ⊔ cdju 7049 |
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 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-coll 4130 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ne 2358 df-ral 2470 df-rex 2471 df-reu 2472 df-rab 2474 df-v 2751 df-sbc 2975 df-csb 3070 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-iun 3900 df-br 4016 df-opab 4077 df-mpt 4078 df-tr 4114 df-id 4305 df-iord 4378 df-on 4380 df-suc 4383 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-fv 5236 df-1st 6154 df-2nd 6155 df-1o 6430 df-er 6548 df-en 6754 df-dju 7050 df-inl 7059 df-inr 7060 |
This theorem is referenced by: enct 12447 |
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