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Theorem tz6.12-1 5344
 Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
tz6.12-1 ((AFB ∃!y AFy) → (FA) = B)
Distinct variable groups:   y,A   y,F
Allowed substitution hint:   B(y)

Proof of Theorem tz6.12-1
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . 3 x AFy
2 nfv 1619 . . 3 y AFx
3 breq2 4643 . . 3 (y = x → (AFyAFx))
41, 2, 3cbveu 2224 . 2 (∃!y AFy∃!x AFx)
5 df-fv 4795 . . 3 (FA) = (℩xAFx)
6 brrelrnex 4691 . . . . 5 (AFBB V)
76adantr 451 . . . 4 ((AFB ∃!x AFx) → B V)
8 breq2 4643 . . . . . . . . 9 (x = B → (AFxAFB))
98iota2 4367 . . . . . . . 8 ((B V ∃!x AFx) → (AFB ↔ (℩xAFx) = B))
109biimpd 198 . . . . . . 7 ((B V ∃!x AFx) → (AFB → (℩xAFx) = B))
1110ex 423 . . . . . 6 (B V → (∃!x AFx → (AFB → (℩xAFx) = B)))
1211com23 72 . . . . 5 (B V → (AFB → (∃!x AFx → (℩xAFx) = B)))
1312imp3a 420 . . . 4 (B V → ((AFB ∃!x AFx) → (℩xAFx) = B))
147, 13mpcom 32 . . 3 ((AFB ∃!x AFx) → (℩xAFx) = B)
155, 14syl5eq 2397 . 2 ((AFB ∃!x AFx) → (FA) = B)
164, 15sylan2b 461 1 ((AFB ∃!y AFy) → (FA) = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859  ℩cio 4337   class class class wbr 4639   ‘cfv 4781 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-br 4640  df-fv 4795 This theorem is referenced by:  tz6.12  5345  tz6.12c  5347  funbrfv  5356  fvfullfunlem3  5863
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