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Mirrors > Home > ILE Home > Th. List > th3qcor | GIF version |
Description: Corollary of Theorem 3Q of [Enderton] p. 60. (Contributed by NM, 12-Nov-1995.) (Revised by David Abernethy, 4-Jun-2013.) |
Ref | Expression |
---|---|
th3q.1 | ⊢ ∼ ∈ V |
th3q.2 | ⊢ ∼ Er (𝑆 × 𝑆) |
th3q.4 | ⊢ ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) ∧ ((𝑠 ∈ 𝑆 ∧ 𝑓 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆))) → ((〈𝑤, 𝑣〉 ∼ 〈𝑢, 𝑡〉 ∧ 〈𝑠, 𝑓〉 ∼ 〈𝑔, ℎ〉) → (〈𝑤, 𝑣〉 + 〈𝑠, 𝑓〉) ∼ (〈𝑢, 𝑡〉 + 〈𝑔, ℎ〉))) |
th3q.5 | ⊢ 𝐺 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [〈𝑤, 𝑣〉] ∼ ∧ 𝑦 = [〈𝑢, 𝑡〉] ∼ ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 + 〈𝑢, 𝑡〉)] ∼ ))} |
Ref | Expression |
---|---|
th3qcor | ⊢ Fun 𝐺 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | th3q.1 | . . . . 5 ⊢ ∼ ∈ V | |
2 | th3q.2 | . . . . 5 ⊢ ∼ Er (𝑆 × 𝑆) | |
3 | th3q.4 | . . . . 5 ⊢ ((((𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑢 ∈ 𝑆 ∧ 𝑡 ∈ 𝑆)) ∧ ((𝑠 ∈ 𝑆 ∧ 𝑓 ∈ 𝑆) ∧ (𝑔 ∈ 𝑆 ∧ ℎ ∈ 𝑆))) → ((〈𝑤, 𝑣〉 ∼ 〈𝑢, 𝑡〉 ∧ 〈𝑠, 𝑓〉 ∼ 〈𝑔, ℎ〉) → (〈𝑤, 𝑣〉 + 〈𝑠, 𝑓〉) ∼ (〈𝑢, 𝑡〉 + 〈𝑔, ℎ〉))) | |
4 | 1, 2, 3 | th3qlem2 6209 | . . . 4 ⊢ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [〈𝑤, 𝑣〉] ∼ ∧ 𝑦 = [〈𝑢, 𝑡〉] ∼ ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 + 〈𝑢, 𝑡〉)] ∼ )) |
5 | moanimv 1975 | . . . 4 ⊢ (∃*𝑧((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [〈𝑤, 𝑣〉] ∼ ∧ 𝑦 = [〈𝑢, 𝑡〉] ∼ ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 + 〈𝑢, 𝑡〉)] ∼ )) ↔ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) → ∃*𝑧∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [〈𝑤, 𝑣〉] ∼ ∧ 𝑦 = [〈𝑢, 𝑡〉] ∼ ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 + 〈𝑢, 𝑡〉)] ∼ ))) | |
6 | 4, 5 | mpbir 134 | . . 3 ⊢ ∃*𝑧((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [〈𝑤, 𝑣〉] ∼ ∧ 𝑦 = [〈𝑢, 𝑡〉] ∼ ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 + 〈𝑢, 𝑡〉)] ∼ )) |
7 | 6 | funoprab 5601 | . 2 ⊢ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [〈𝑤, 𝑣〉] ∼ ∧ 𝑦 = [〈𝑢, 𝑡〉] ∼ ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 + 〈𝑢, 𝑡〉)] ∼ ))} |
8 | th3q.5 | . . 3 ⊢ 𝐺 = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [〈𝑤, 𝑣〉] ∼ ∧ 𝑦 = [〈𝑢, 𝑡〉] ∼ ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 + 〈𝑢, 𝑡〉)] ∼ ))} | |
9 | 8 | funeqi 4922 | . 2 ⊢ (Fun 𝐺 ↔ Fun {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ((𝑆 × 𝑆) / ∼ ) ∧ 𝑦 ∈ ((𝑆 × 𝑆) / ∼ )) ∧ ∃𝑤∃𝑣∃𝑢∃𝑡((𝑥 = [〈𝑤, 𝑣〉] ∼ ∧ 𝑦 = [〈𝑢, 𝑡〉] ∼ ) ∧ 𝑧 = [(〈𝑤, 𝑣〉 + 〈𝑢, 𝑡〉)] ∼ ))}) |
10 | 7, 9 | mpbir 134 | 1 ⊢ Fun 𝐺 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∃wex 1381 ∈ wcel 1393 ∃*wmo 1901 Vcvv 2557 〈cop 3378 class class class wbr 3764 × cxp 4343 Fun wfun 4896 (class class class)co 5512 {coprab 5513 Er wer 6103 [cec 6104 / cqs 6105 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-sbc 2765 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fv 4910 df-ov 5515 df-oprab 5516 df-er 6106 df-ec 6108 df-qs 6112 |
This theorem is referenced by: (None) |
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