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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 6694 | . . . 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 2117 | . . . 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 146 | . . 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 6019 | . 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 5276 | . 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 146 | 1 ⊢ Fun 𝐺 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1364 ∃wex 1503 ∃*wmo 2043 ∈ wcel 2164 Vcvv 2760 〈cop 3622 class class class wbr 4030 × cxp 4658 Fun wfun 5249 (class class class)co 5919 {coprab 5920 Er wer 6586 [cec 6587 / cqs 6588 |
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-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-pr 4239 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-sbc 2987 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-br 4031 df-opab 4092 df-id 4325 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fv 5263 df-ov 5922 df-oprab 5923 df-er 6589 df-ec 6591 df-qs 6595 |
This theorem is referenced by: (None) |
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