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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 6614 | . . . 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 2094 | . . . 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 5951 | . 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 5217 | . 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 1348 ∃wex 1485 ∃*wmo 2020 ∈ wcel 2141 Vcvv 2730 〈cop 3584 class class class wbr 3987 × cxp 4607 Fun wfun 5190 (class class class)co 5851 {coprab 5852 Er wer 6508 [cec 6509 / cqs 6510 |
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 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-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 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-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fv 5204 df-ov 5854 df-oprab 5855 df-er 6511 df-ec 6513 df-qs 6517 |
This theorem is referenced by: (None) |
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