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Theorem fun11 5148
Description: Two ways of stating that 𝐴 is one-to-one (but not necessarily a function). Each side is equivalent to Definition 6.4(3) of [TakeutiZaring] p. 24, who use the notation "Un2 (A)" for one-to-one (but not necessarily a function). (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
fun11 ((Fun 𝐴 ∧ Fun 𝐴) ↔ ∀𝑥𝑦𝑧𝑤((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤,𝐴

Proof of Theorem fun11
StepHypRef Expression
1 dfbi2 383 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) ↔ ((𝑥 = 𝑧𝑦 = 𝑤) ∧ (𝑦 = 𝑤𝑥 = 𝑧)))
21imbi2i 225 . . . . . . 7 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝑥𝐴𝑦𝑧𝐴𝑤) → ((𝑥 = 𝑧𝑦 = 𝑤) ∧ (𝑦 = 𝑤𝑥 = 𝑧))))
3 pm4.76 576 . . . . . . 7 ((((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ∧ ((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑦 = 𝑤𝑥 = 𝑧))) ↔ ((𝑥𝐴𝑦𝑧𝐴𝑤) → ((𝑥 = 𝑧𝑦 = 𝑤) ∧ (𝑦 = 𝑤𝑥 = 𝑧))))
4 bi2.04 247 . . . . . . . 8 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)))
5 bi2.04 247 . . . . . . . 8 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑦 = 𝑤𝑥 = 𝑧)) ↔ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧)))
64, 5anbi12i 453 . . . . . . 7 ((((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ∧ ((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑦 = 𝑤𝑥 = 𝑧))) ↔ ((𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))))
72, 3, 63bitr2i 207 . . . . . 6 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))))
872albii 1430 . . . . 5 (∀𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦((𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))))
9 19.26-2 1441 . . . . 5 (∀𝑥𝑦((𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))) ↔ (∀𝑥𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ ∀𝑥𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))))
10 alcom 1437 . . . . . . 7 (∀𝑥𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)))
11 nfv 1491 . . . . . . . . 9 𝑥((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)
12 breq1 3898 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥𝐴𝑦𝑧𝐴𝑦))
1312anbi1d 458 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) ↔ (𝑧𝐴𝑦𝑧𝐴𝑤)))
1413imbi1d 230 . . . . . . . . 9 (𝑥 = 𝑧 → (((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)))
1511, 14equsal 1688 . . . . . . . 8 (∀𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
1615albii 1429 . . . . . . 7 (∀𝑦𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
1710, 16bitri 183 . . . . . 6 (∀𝑥𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
18 nfv 1491 . . . . . . . 8 𝑦((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)
19 breq2 3899 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑥𝐴𝑦𝑥𝐴𝑤))
2019anbi1d 458 . . . . . . . . 9 (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) ↔ (𝑥𝐴𝑤𝑧𝐴𝑤)))
2120imbi1d 230 . . . . . . . 8 (𝑦 = 𝑤 → (((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
2218, 21equsal 1688 . . . . . . 7 (∀𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
2322albii 1429 . . . . . 6 (∀𝑥𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
2417, 23anbi12i 453 . . . . 5 ((∀𝑥𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ ∀𝑥𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))) ↔ (∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
258, 9, 243bitri 205 . . . 4 (∀𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
26252albii 1430 . . 3 (∀𝑧𝑤𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑧𝑤(∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
27 19.26-2 1441 . . 3 (∀𝑧𝑤(∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ (∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
2826, 27bitr2i 184 . 2 ((∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ∀𝑧𝑤𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))
29 fun2cnv 5145 . . . 4 (Fun 𝐴 ↔ ∀𝑧∃*𝑦 𝑧𝐴𝑦)
30 breq2 3899 . . . . . 6 (𝑦 = 𝑤 → (𝑧𝐴𝑦𝑧𝐴𝑤))
3130mo4 2036 . . . . 5 (∃*𝑦 𝑧𝐴𝑦 ↔ ∀𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
3231albii 1429 . . . 4 (∀𝑧∃*𝑦 𝑧𝐴𝑦 ↔ ∀𝑧𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
33 alcom 1437 . . . . 5 (∀𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ∀𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
3433albii 1429 . . . 4 (∀𝑧𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
3529, 32, 343bitri 205 . . 3 (Fun 𝐴 ↔ ∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
36 funcnv2 5141 . . . 4 (Fun 𝐴 ↔ ∀𝑤∃*𝑥 𝑥𝐴𝑤)
37 breq1 3898 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑤𝑧𝐴𝑤))
3837mo4 2036 . . . . 5 (∃*𝑥 𝑥𝐴𝑤 ↔ ∀𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
3938albii 1429 . . . 4 (∀𝑤∃*𝑥 𝑥𝐴𝑤 ↔ ∀𝑤𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
40 alcom 1437 . . . . . 6 (∀𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
4140albii 1429 . . . . 5 (∀𝑤𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑤𝑧𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
42 alcom 1437 . . . . 5 (∀𝑤𝑧𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
4341, 42bitri 183 . . . 4 (∀𝑤𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
4436, 39, 433bitri 205 . . 3 (Fun 𝐴 ↔ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
4535, 44anbi12i 453 . 2 ((Fun 𝐴 ∧ Fun 𝐴) ↔ (∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
46 alrot4 1445 . 2 (∀𝑥𝑦𝑧𝑤((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑧𝑤𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))
4728, 45, 463bitr4i 211 1 ((Fun 𝐴 ∧ Fun 𝐴) ↔ ∀𝑥𝑦𝑧𝑤((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1312  ∃*wmo 1976   class class class wbr 3895  ccnv 4498  Fun wfun 5075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-fun 5083
This theorem is referenced by: (None)
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