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Theorem fun11 6174
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 467 . . . . . . . 8 ((𝑥 = 𝑧𝑦 = 𝑤) ↔ ((𝑥 = 𝑧𝑦 = 𝑤) ∧ (𝑦 = 𝑤𝑥 = 𝑧)))
21imbi2i 328 . . . . . . 7 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝑥𝐴𝑦𝑧𝐴𝑤) → ((𝑥 = 𝑧𝑦 = 𝑤) ∧ (𝑦 = 𝑤𝑥 = 𝑧))))
3 pm4.76 515 . . . . . . 7 ((((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ∧ ((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑦 = 𝑤𝑥 = 𝑧))) ↔ ((𝑥𝐴𝑦𝑧𝐴𝑤) → ((𝑥 = 𝑧𝑦 = 𝑤) ∧ (𝑦 = 𝑤𝑥 = 𝑧))))
4 bi2.04 378 . . . . . . . 8 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)))
5 bi2.04 378 . . . . . . . 8 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑦 = 𝑤𝑥 = 𝑧)) ↔ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧)))
64, 5anbi12i 621 . . . . . . 7 ((((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ∧ ((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑦 = 𝑤𝑥 = 𝑧))) ↔ ((𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))))
72, 3, 63bitr2i 291 . . . . . 6 (((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ((𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))))
872albii 1916 . . . . 5 (∀𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑥𝑦((𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))))
9 19.26-2 1970 . . . . 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 2202 . . . . . . 7 (∀𝑥𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)))
11 breq1 4846 . . . . . . . . . . 11 (𝑥 = 𝑧 → (𝑥𝐴𝑦𝑧𝐴𝑦))
1211anbi1d 624 . . . . . . . . . 10 (𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) ↔ (𝑧𝐴𝑦𝑧𝐴𝑤)))
1312imbi1d 333 . . . . . . . . 9 (𝑥 = 𝑧 → (((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)))
1413equsalvw 2103 . . . . . . . 8 (∀𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
1514albii 1915 . . . . . . 7 (∀𝑦𝑥(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
1610, 15bitri 267 . . . . . 6 (∀𝑥𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ↔ ∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
17 breq2 4847 . . . . . . . . . 10 (𝑦 = 𝑤 → (𝑥𝐴𝑦𝑥𝐴𝑤))
1817anbi1d 624 . . . . . . . . 9 (𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) ↔ (𝑥𝐴𝑤𝑧𝐴𝑤)))
1918imbi1d 333 . . . . . . . 8 (𝑦 = 𝑤 → (((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
2019equsalvw 2103 . . . . . . 7 (∀𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
2120albii 1915 . . . . . 6 (∀𝑥𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
2216, 21anbi12i 621 . . . . 5 ((∀𝑥𝑦(𝑥 = 𝑧 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤)) ∧ ∀𝑥𝑦(𝑦 = 𝑤 → ((𝑥𝐴𝑦𝑧𝐴𝑤) → 𝑥 = 𝑧))) ↔ (∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
238, 9, 223bitri 289 . . . 4 (∀𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ (∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
24232albii 1916 . . 3 (∀𝑧𝑤𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑧𝑤(∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
25 19.26-2 1970 . . 3 (∀𝑧𝑤(∀𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ (∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
2624, 25bitr2i 268 . 2 ((∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)) ↔ ∀𝑧𝑤𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))
27 fun2cnv 6171 . . . 4 (Fun 𝐴 ↔ ∀𝑧∃*𝑦 𝑧𝐴𝑦)
28 breq2 4847 . . . . . 6 (𝑦 = 𝑤 → (𝑧𝐴𝑦𝑧𝐴𝑤))
2928mo4 2671 . . . . 5 (∃*𝑦 𝑧𝐴𝑦 ↔ ∀𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
3029albii 1915 . . . 4 (∀𝑧∃*𝑦 𝑧𝐴𝑦 ↔ ∀𝑧𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
31 alcom 2202 . . . . 5 (∀𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ∀𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
3231albii 1915 . . . 4 (∀𝑧𝑦𝑤((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ↔ ∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
3327, 30, 323bitri 289 . . 3 (Fun 𝐴 ↔ ∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤))
34 funcnv2 6168 . . . 4 (Fun 𝐴 ↔ ∀𝑤∃*𝑥 𝑥𝐴𝑤)
35 breq1 4846 . . . . . 6 (𝑥 = 𝑧 → (𝑥𝐴𝑤𝑧𝐴𝑤))
3635mo4 2671 . . . . 5 (∃*𝑥 𝑥𝐴𝑤 ↔ ∀𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
3736albii 1915 . . . 4 (∀𝑤∃*𝑥 𝑥𝐴𝑤 ↔ ∀𝑤𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
38 alcom 2202 . . . . . 6 (∀𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
3938albii 1915 . . . . 5 (∀𝑤𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑤𝑧𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
40 alcom 2202 . . . . 5 (∀𝑤𝑧𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
4139, 40bitri 267 . . . 4 (∀𝑤𝑥𝑧((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧) ↔ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
4234, 37, 413bitri 289 . . 3 (Fun 𝐴 ↔ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧))
4333, 42anbi12i 621 . 2 ((Fun 𝐴 ∧ Fun 𝐴) ↔ (∀𝑧𝑤𝑦((𝑧𝐴𝑦𝑧𝐴𝑤) → 𝑦 = 𝑤) ∧ ∀𝑧𝑤𝑥((𝑥𝐴𝑤𝑧𝐴𝑤) → 𝑥 = 𝑧)))
44 alrot4 2204 . 2 (∀𝑥𝑦𝑧𝑤((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)) ↔ ∀𝑧𝑤𝑥𝑦((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))
4526, 43, 443bitr4i 295 1 ((Fun 𝐴 ∧ Fun 𝐴) ↔ ∀𝑥𝑦𝑧𝑤((𝑥𝐴𝑦𝑧𝐴𝑤) → (𝑥 = 𝑧𝑦 = 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wal 1651  ∃*wmo 2589   class class class wbr 4843  ccnv 5311  Fun wfun 6095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-fun 6103
This theorem is referenced by: (None)
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