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Theorem funsi 5520
 Description: The singleton image of a function is a function. (Contributed by SF, 26-Feb-2015.)
Assertion
Ref Expression
funsi (Fun F → Fun SI F)

Proof of Theorem funsi
Dummy variables a b x y c d z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brsi 4761 . . . . . . 7 (x SI Fyab(x = {a} y = {b} aFb))
2 brsi 4761 . . . . . . 7 (x SI Fzcd(x = {c} z = {d} cFd))
31, 2anbi12i 678 . . . . . 6 ((x SI Fy x SI Fz) ↔ (ab(x = {a} y = {b} aFb) cd(x = {c} z = {d} cFd)))
4 ee4anv 1915 . . . . . 6 (abcd((x = {a} y = {b} aFb) (x = {c} z = {d} cFd)) ↔ (ab(x = {a} y = {b} aFb) cd(x = {c} z = {d} cFd)))
53, 4bitr4i 243 . . . . 5 ((x SI Fy x SI Fz) ↔ abcd((x = {a} y = {b} aFb) (x = {c} z = {d} cFd)))
6 fununiq 5517 . . . . . . . . . . 11 ((Fun F aFb aFd) → b = d)
763exp 1150 . . . . . . . . . 10 (Fun F → (aFb → (aFdb = d)))
8 breq1 4642 . . . . . . . . . . . . . . . 16 (a = c → (aFdcFd))
98bicomd 192 . . . . . . . . . . . . . . 15 (a = c → (cFdaFd))
109adantr 451 . . . . . . . . . . . . . 14 ((a = c z = {d}) → (cFdaFd))
11 eqeq2 2362 . . . . . . . . . . . . . . . 16 (z = {d} → ({b} = z ↔ {b} = {d}))
12 vex 2862 . . . . . . . . . . . . . . . . 17 b V
1312sneqb 3876 . . . . . . . . . . . . . . . 16 ({b} = {d} ↔ b = d)
1411, 13syl6bb 252 . . . . . . . . . . . . . . 15 (z = {d} → ({b} = zb = d))
1514adantl 452 . . . . . . . . . . . . . 14 ((a = c z = {d}) → ({b} = zb = d))
1610, 15imbi12d 311 . . . . . . . . . . . . 13 ((a = c z = {d}) → ((cFd → {b} = z) ↔ (aFdb = d)))
1716biimprcd 216 . . . . . . . . . . . 12 ((aFdb = d) → ((a = c z = {d}) → (cFd → {b} = z)))
1817exp3a 425 . . . . . . . . . . 11 ((aFdb = d) → (a = c → (z = {d} → (cFd → {b} = z))))
19183impd 1165 . . . . . . . . . 10 ((aFdb = d) → ((a = c z = {d} cFd) → {b} = z))
207, 19syl6 29 . . . . . . . . 9 (Fun F → (aFb → ((a = c z = {d} cFd) → {b} = z)))
21 eqeq1 2359 . . . . . . . . . . . . . . . . 17 (x = {a} → (x = {c} ↔ {a} = {c}))
22 vex 2862 . . . . . . . . . . . . . . . . . 18 a V
2322sneqb 3876 . . . . . . . . . . . . . . . . 17 ({a} = {c} ↔ a = c)
2421, 23syl6bb 252 . . . . . . . . . . . . . . . 16 (x = {a} → (x = {c} ↔ a = c))
25243anbi1d 1256 . . . . . . . . . . . . . . 15 (x = {a} → ((x = {c} z = {d} cFd) ↔ (a = c z = {d} cFd)))
2625adantr 451 . . . . . . . . . . . . . 14 ((x = {a} y = {b}) → ((x = {c} z = {d} cFd) ↔ (a = c z = {d} cFd)))
27 eqeq1 2359 . . . . . . . . . . . . . . 15 (y = {b} → (y = z ↔ {b} = z))
2827adantl 452 . . . . . . . . . . . . . 14 ((x = {a} y = {b}) → (y = z ↔ {b} = z))
2926, 28imbi12d 311 . . . . . . . . . . . . 13 ((x = {a} y = {b}) → (((x = {c} z = {d} cFd) → y = z) ↔ ((a = c z = {d} cFd) → {b} = z)))
3029imbi2d 307 . . . . . . . . . . . 12 ((x = {a} y = {b}) → ((aFb → ((x = {c} z = {d} cFd) → y = z)) ↔ (aFb → ((a = c z = {d} cFd) → {b} = z))))
3130biimprcd 216 . . . . . . . . . . 11 ((aFb → ((a = c z = {d} cFd) → {b} = z)) → ((x = {a} y = {b}) → (aFb → ((x = {c} z = {d} cFd) → y = z))))
3231exp3a 425 . . . . . . . . . 10 ((aFb → ((a = c z = {d} cFd) → {b} = z)) → (x = {a} → (y = {b} → (aFb → ((x = {c} z = {d} cFd) → y = z)))))
33323impd 1165 . . . . . . . . 9 ((aFb → ((a = c z = {d} cFd) → {b} = z)) → ((x = {a} y = {b} aFb) → ((x = {c} z = {d} cFd) → y = z)))
3420, 33syl 15 . . . . . . . 8 (Fun F → ((x = {a} y = {b} aFb) → ((x = {c} z = {d} cFd) → y = z)))
3534imp3a 420 . . . . . . 7 (Fun F → (((x = {a} y = {b} aFb) (x = {c} z = {d} cFd)) → y = z))
3635exlimdvv 1637 . . . . . 6 (Fun F → (cd((x = {a} y = {b} aFb) (x = {c} z = {d} cFd)) → y = z))
3736exlimdvv 1637 . . . . 5 (Fun F → (abcd((x = {a} y = {b} aFb) (x = {c} z = {d} cFd)) → y = z))
385, 37syl5bi 208 . . . 4 (Fun F → ((x SI Fy x SI Fz) → y = z))
3938alrimiv 1631 . . 3 (Fun Fz((x SI Fy x SI Fz) → y = z))
4039alrimivv 1632 . 2 (Fun Fxyz((x SI Fy x SI Fz) → y = z))
41 dffun2 5119 . 2 (Fun SI Fxyz((x SI Fy x SI Fz) → y = z))
4240, 41sylibr 203 1 (Fun F → Fun SI F)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∀wal 1540  ∃wex 1541   = wceq 1642  {csn 3737   class class class wbr 4639   SI csi 4720  Fun wfun 4775 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-co 4726  df-si 4728  df-id 4767  df-cnv 4785  df-fun 4789 This theorem is referenced by:  enpw1  6062
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