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Theorem fun11 5275
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 388 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) ↔ ((𝑥 = 𝑧𝑦 = 𝑤) ∧ (𝑦 = 𝑤𝑥 = 𝑧)))
21imbi2i 226 . . . . . . 7 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝑥𝐴𝑦𝑧𝐴𝑤) → ((𝑥 = 𝑧𝑦 = 𝑤) ∧ (𝑦 = 𝑤𝑥 = 𝑧))))
3 pm4.76 604 . . . . . . 7 ((((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ∧ ((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑦 = 𝑤𝑥 = 𝑧))) ↔ ((𝑥𝐴𝑦𝑧𝐴𝑤) → ((𝑥 = 𝑧𝑦 = 𝑤) ∧ (𝑦 = 𝑤𝑥 = 𝑧))))
4 bi2.04 248 . . . . . . . 8 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)))
5 bi2.04 248 . . . . . . . 8 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑦 = 𝑤𝑥 = 𝑧)) ↔ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧)))
64, 5anbi12i 460 . . . . . . 7 ((((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ∧ ((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑦 = 𝑤𝑥 = 𝑧))) ↔ ((𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))))
72, 3, 63bitr2i 208 . . . . . 6 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))))
872albii 1469 . . . . 5 (∀𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦((𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))))
9 19.26-2 1480 . . . . 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 1476 . . . . . . 7 (∀𝑥𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)))
11 nfv 1526 . . . . . . . . 9 𝑥((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)
12 breq1 4001 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥𝐴𝑦𝑧𝐴𝑦))
1312anbi1d 465 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) ↔ (𝑧𝐴𝑦𝑧𝐴𝑤)))
1413imbi1d 231 . . . . . . . . 9 (𝑥 = 𝑧 → (((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)))
1511, 14equsal 1725 . . . . . . . 8 (∀𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
1615albii 1468 . . . . . . 7 (∀𝑦𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
1710, 16bitri 184 . . . . . 6 (∀𝑥𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
18 nfv 1526 . . . . . . . 8 𝑦((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)
19 breq2 4002 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑥𝐴𝑦𝑥𝐴𝑤))
2019anbi1d 465 . . . . . . . . 9 (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) ↔ (𝑥𝐴𝑤𝑧𝐴𝑤)))
2120imbi1d 231 . . . . . . . 8 (𝑦 = 𝑤 → (((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
2218, 21equsal 1725 . . . . . . 7 (∀𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
2322albii 1468 . . . . . 6 (∀𝑥𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
2417, 23anbi12i 460 . . . . 5 ((∀𝑥𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ ∀𝑥𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))) ↔ (∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
258, 9, 243bitri 206 . . . 4 (∀𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
26252albii 1469 . . 3 (∀𝑧𝑤𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑧𝑤(∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
27 19.26-2 1480 . . 3 (∀𝑧𝑤(∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ (∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
2826, 27bitr2i 185 . 2 ((∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ∀𝑧𝑤𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))
29 fun2cnv 5272 . . . 4 (Fun 𝐴 ↔ ∀𝑧∃*𝑦 𝑧𝐴𝑦)
30 breq2 4002 . . . . . 6 (𝑦 = 𝑤 → (𝑧𝐴𝑦𝑧𝐴𝑤))
3130mo4 2085 . . . . 5 (∃*𝑦 𝑧𝐴𝑦 ↔ ∀𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
3231albii 1468 . . . 4 (∀𝑧∃*𝑦 𝑧𝐴𝑦 ↔ ∀𝑧𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
33 alcom 1476 . . . . 5 (∀𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ∀𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
3433albii 1468 . . . 4 (∀𝑧𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
3529, 32, 343bitri 206 . . 3 (Fun 𝐴 ↔ ∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
36 funcnv2 5268 . . . 4 (Fun 𝐴 ↔ ∀𝑤∃*𝑥 𝑥𝐴𝑤)
37 breq1 4001 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑤𝑧𝐴𝑤))
3837mo4 2085 . . . . 5 (∃*𝑥 𝑥𝐴𝑤 ↔ ∀𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
3938albii 1468 . . . 4 (∀𝑤∃*𝑥 𝑥𝐴𝑤 ↔ ∀𝑤𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
40 alcom 1476 . . . . . 6 (∀𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
4140albii 1468 . . . . 5 (∀𝑤𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑤𝑧𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
42 alcom 1476 . . . . 5 (∀𝑤𝑧𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
4341, 42bitri 184 . . . 4 (∀𝑤𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
4436, 39, 433bitri 206 . . 3 (Fun 𝐴 ↔ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
4535, 44anbi12i 460 . 2 ((Fun 𝐴 ∧ Fun 𝐴) ↔ (∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
46 alrot4 1484 . 2 (∀𝑥𝑦𝑧𝑤((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑧𝑤𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))
4728, 45, 463bitr4i 212 1 ((Fun 𝐴 ∧ Fun 𝐴) ↔ ∀𝑥𝑦𝑧𝑤((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351  ∃*wmo 2025   class class class wbr 3998  ccnv 4619  Fun wfun 5202
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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-fun 5210
This theorem is referenced by: (None)
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