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Theorem fundmen 6043
Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by SF, 23-Feb-2015.)
Hypothesis
Ref Expression
fundmen.1 F V
Assertion
Ref Expression
fundmen (Fun F → dom FF)

Proof of Theorem fundmen
Dummy variables x y z a b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssv 3291 . . . . . 6 F V
2 1stfo 5505 . . . . . . 7 1st :V–onto→V
3 fofn 5271 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
4 fnssresb 5195 . . . . . . 7 (1st Fn V → ((1st F) Fn FF V))
52, 3, 4mp2b 9 . . . . . 6 ((1st F) Fn FF V)
61, 5mpbir 200 . . . . 5 (1st F) Fn F
76a1i 10 . . . 4 (Fun F → (1st F) Fn F)
8 brcnv 4892 . . . . . . . . . . 11 (x(1st F)yy(1st F)x)
9 brres 4949 . . . . . . . . . . 11 (y(1st F)x ↔ (y1st x y F))
10 vex 2862 . . . . . . . . . . . . . 14 x V
1110br1st 4858 . . . . . . . . . . . . 13 (y1st xa y = x, a)
1211anbi1i 676 . . . . . . . . . . . 12 ((y1st x y F) ↔ (a y = x, a y F))
13 19.41v 1901 . . . . . . . . . . . 12 (a(y = x, a y F) ↔ (a y = x, a y F))
1412, 13bitr4i 243 . . . . . . . . . . 11 ((y1st x y F) ↔ a(y = x, a y F))
158, 9, 143bitri 262 . . . . . . . . . 10 (x(1st F)ya(y = x, a y F))
16 brcnv 4892 . . . . . . . . . . . 12 (x(1st F)zz(1st F)x)
17 brres 4949 . . . . . . . . . . . 12 (z(1st F)x ↔ (z1st x z F))
1810br1st 4858 . . . . . . . . . . . . 13 (z1st xb z = x, b)
1918anbi1i 676 . . . . . . . . . . . 12 ((z1st x z F) ↔ (b z = x, b z F))
2016, 17, 193bitri 262 . . . . . . . . . . 11 (x(1st F)z ↔ (b z = x, b z F))
21 19.41v 1901 . . . . . . . . . . 11 (b(z = x, b z F) ↔ (b z = x, b z F))
2220, 21bitr4i 243 . . . . . . . . . 10 (x(1st F)zb(z = x, b z F))
2315, 22anbi12i 678 . . . . . . . . 9 ((x(1st F)y x(1st F)z) ↔ (a(y = x, a y F) b(z = x, b z F)))
24 eeanv 1913 . . . . . . . . 9 (ab((y = x, a y F) (z = x, b z F)) ↔ (a(y = x, a y F) b(z = x, b z F)))
2523, 24bitr4i 243 . . . . . . . 8 ((x(1st F)y x(1st F)z) ↔ ab((y = x, a y F) (z = x, b z F)))
26 an4 797 . . . . . . . . . 10 (((y = x, a y F) (z = x, b z F)) ↔ ((y = x, a z = x, b) (y F z F)))
27 dffun4 5121 . . . . . . . . . . . . 13 (Fun Fxab((x, a F x, b F) → a = b))
28 sp 1747 . . . . . . . . . . . . . . 15 (b((x, a F x, b F) → a = b) → ((x, a F x, b F) → a = b))
2928sps 1754 . . . . . . . . . . . . . 14 (ab((x, a F x, b F) → a = b) → ((x, a F x, b F) → a = b))
3029sps 1754 . . . . . . . . . . . . 13 (xab((x, a F x, b F) → a = b) → ((x, a F x, b F) → a = b))
3127, 30sylbi 187 . . . . . . . . . . . 12 (Fun F → ((x, a F x, b F) → a = b))
32 opeq2 4579 . . . . . . . . . . . 12 (a = bx, a = x, b)
3331, 32syl6 29 . . . . . . . . . . 11 (Fun F → ((x, a F x, b F) → x, a = x, b))
34 eleq1 2413 . . . . . . . . . . . . . . 15 (y = x, a → (y Fx, a F))
35 eleq1 2413 . . . . . . . . . . . . . . 15 (z = x, b → (z Fx, b F))
3634, 35bi2anan9 843 . . . . . . . . . . . . . 14 ((y = x, a z = x, b) → ((y F z F) ↔ (x, a F x, b F)))
37 eqeq12 2365 . . . . . . . . . . . . . 14 ((y = x, a z = x, b) → (y = zx, a = x, b))
3836, 37imbi12d 311 . . . . . . . . . . . . 13 ((y = x, a z = x, b) → (((y F z F) → y = z) ↔ ((x, a F x, b F) → x, a = x, b)))
3938biimprcd 216 . . . . . . . . . . . 12 (((x, a F x, b F) → x, a = x, b) → ((y = x, a z = x, b) → ((y F z F) → y = z)))
4039imp3a 420 . . . . . . . . . . 11 (((x, a F x, b F) → x, a = x, b) → (((y = x, a z = x, b) (y F z F)) → y = z))
4133, 40syl 15 . . . . . . . . . 10 (Fun F → (((y = x, a z = x, b) (y F z F)) → y = z))
4226, 41syl5bi 208 . . . . . . . . 9 (Fun F → (((y = x, a y F) (z = x, b z F)) → y = z))
4342exlimdvv 1637 . . . . . . . 8 (Fun F → (ab((y = x, a y F) (z = x, b z F)) → y = z))
4425, 43syl5bi 208 . . . . . . 7 (Fun F → ((x(1st F)y x(1st F)z) → y = z))
4544alrimiv 1631 . . . . . 6 (Fun Fz((x(1st F)y x(1st F)z) → y = z))
4645alrimivv 1632 . . . . 5 (Fun Fxyz((x(1st F)y x(1st F)z) → y = z))
47 dffun2 5119 . . . . 5 (Fun (1st F) ↔ xyz((x(1st F)y x(1st F)z) → y = z))
4846, 47sylibr 203 . . . 4 (Fun F → Fun (1st F))
49 dfdm4 5507 . . . . . 6 dom F = (1stF)
50 dfima3 4951 . . . . . 6 (1stF) = ran (1st F)
5149, 50eqtr2i 2374 . . . . 5 ran (1st F) = dom F
5251a1i 10 . . . 4 (Fun F → ran (1st F) = dom F)
53 dff1o2 5291 . . . 4 ((1st F):F1-1-onto→dom F ↔ ((1st F) Fn F Fun (1st F) ran (1st F) = dom F))
547, 48, 52, 53syl3anbrc 1136 . . 3 (Fun F → (1st F):F1-1-onto→dom F)
55 1stex 4739 . . . . 5 1st V
56 fundmen.1 . . . . 5 F V
5755, 56resex 5117 . . . 4 (1st F) V
5857f1oen 6033 . . 3 ((1st F):F1-1-onto→dom FF ≈ dom F)
5954, 58syl 15 . 2 (Fun FF ≈ dom F)
60 ensym 6037 . 2 (F ≈ dom F ↔ dom FF)
6159, 60sylib 188 1 (Fun F → dom FF)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 176   wa 358  wal 1540  wex 1541   = wceq 1642   wcel 1710  Vcvv 2859   wss 3257  cop 4561   class class class wbr 4639  1st c1st 4717  cima 4722  ccnv 4771  dom cdm 4772  ran crn 4773   cres 4774  Fun wfun 4775   Fn wfn 4776  ontowfo 4779  1-1-ontowf1o 4780  cen 6028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-en 6029
This theorem is referenced by:  fundmeng  6044  xpsnen  6049
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