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Mirrors > Home > ILE Home > Th. List > enctlem | GIF version |
Description: Lemma for enct 11982. 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 6329 | . . . . 5 ⊢ 1o ∈ V | |
2 | 1 | enref 6667 | . . . 4 ⊢ 1o ≈ 1o |
3 | djuen 7084 | . . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 1o ≈ 1o) → (𝐴 ⊔ 1o) ≈ (𝐵 ⊔ 1o)) | |
4 | 2, 3 | mpan2 422 | . . 3 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ⊔ 1o) ≈ (𝐵 ⊔ 1o)) |
5 | bren 6649 | . . 3 ⊢ ((𝐴 ⊔ 1o) ≈ (𝐵 ⊔ 1o) ↔ ∃ℎ ℎ:(𝐴 ⊔ 1o)–1-1-onto→(𝐵 ⊔ 1o)) | |
6 | 4, 5 | sylib 121 | . 2 ⊢ (𝐴 ≈ 𝐵 → ∃ℎ ℎ:(𝐴 ⊔ 1o)–1-1-onto→(𝐵 ⊔ 1o)) |
7 | f1ofo 5382 | . . . . . 6 ⊢ (ℎ:(𝐴 ⊔ 1o)–1-1-onto→(𝐵 ⊔ 1o) → ℎ:(𝐴 ⊔ 1o)–onto→(𝐵 ⊔ 1o)) | |
8 | 7 | ad2antlr 481 | . . . . 5 ⊢ (((𝐴 ≈ 𝐵 ∧ ℎ:(𝐴 ⊔ 1o)–1-1-onto→(𝐵 ⊔ 1o)) ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → ℎ:(𝐴 ⊔ 1o)–onto→(𝐵 ⊔ 1o)) |
9 | foco 5363 | . . . . . 6 ⊢ ((ℎ:(𝐴 ⊔ 1o)–onto→(𝐵 ⊔ 1o) ∧ 𝑓:ω–onto→(𝐴 ⊔ 1o)) → (ℎ ∘ 𝑓):ω–onto→(𝐵 ⊔ 1o)) | |
10 | vex 2692 | . . . . . . . 8 ⊢ ℎ ∈ V | |
11 | vex 2692 | . . . . . . . 8 ⊢ 𝑓 ∈ V | |
12 | 10, 11 | coex 5092 | . . . . . . 7 ⊢ (ℎ ∘ 𝑓) ∈ V |
13 | foeq1 5349 | . . . . . . 7 ⊢ (𝑔 = (ℎ ∘ 𝑓) → (𝑔:ω–onto→(𝐵 ⊔ 1o) ↔ (ℎ ∘ 𝑓):ω–onto→(𝐵 ⊔ 1o))) | |
14 | 12, 13 | spcev 2784 | . . . . . 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 417 | . . . 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 114 | . . 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 1792 | . 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 1871 | 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 103 ∃wex 1469 class class class wbr 3937 ωcom 4512 ∘ ccom 4551 –onto→wfo 5129 –1-1-onto→wf1o 5130 1oc1o 6314 ≈ cen 6640 ⊔ cdju 6930 |
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-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 |
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-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 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-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1st 6046 df-2nd 6047 df-1o 6321 df-er 6437 df-en 6643 df-dju 6931 df-inl 6940 df-inr 6941 |
This theorem is referenced by: enct 11982 |
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