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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 6532 | . . . 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 2074 | . . . 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 145 | . . 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 5871 | . 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 5144 | . 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 145 | 1 ⊢ Fun 𝐺 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∃*wmo 2000 Vcvv 2686 〈cop 3530 class class class wbr 3929 × cxp 4537 Fun wfun 5117 (class class class)co 5774 {coprab 5775 Er wer 6426 [cec 6427 / cqs 6428 |
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-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-er 6429 df-ec 6431 df-qs 6435 |
This theorem is referenced by: (None) |
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