Theorem tz6.12-1 5345

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) → (F ‘A) = 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.

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 ⊢ Ⅎx AFy
2		nfv 1619	. . 3 ⊢ Ⅎy AFx
3		breq2 4644	. . 3 ⊢ (y = x → (AFy ↔ AFx))
4	1, 2, 3	cbveu 2224	. 2 ⊢ (∃!y AFy ↔ ∃!x AFx)
5		df-fv 4796	. . 3 ⊢ (F ‘A) = (℩xAFx)
6		brrelrnex 4692	. . . . 5 ⊢ (AFB → B ∈ V)
7	6	adantr 451	. . . 4 ⊢ ((AFB ∧ ∃!x AFx) → B ∈ V)
8		breq2 4644	. . . . . . . . 9 ⊢ (x = B → (AFx ↔ AFB))
9	8	iota2 4368	. . . . . . . 8 ⊢ ((B ∈ V ∧ ∃!x AFx) → (AFB ↔ (℩xAFx) = B))
10	9	biimpd 198	. . . . . . 7 ⊢ ((B ∈ V ∧ ∃!x AFx) → (AFB → (℩xAFx) = B))
11	10	ex 423	. . . . . 6 ⊢ (B ∈ V → (∃!x AFx → (AFB → (℩xAFx) = B)))
12	11	com23 72	. . . . 5 ⊢ (B ∈ V → (AFB → (∃!x AFx → (℩xAFx) = B)))
13	12	imp3a 420	. . . 4 ⊢ (B ∈ V → ((AFB ∧ ∃!x AFx) → (℩xAFx) = B))
14	7, 13	mpcom 32	. . 3 ⊢ ((AFB ∧ ∃!x AFx) → (℩xAFx) = B)
15	5, 14	syl5eq 2397	. 2 ⊢ ((AFB ∧ ∃!x AFx) → (F ‘A) = B)
16	4, 15	sylan2b 461	1 ⊢ ((AFB ∧ ∃!y AFy) → (F ‘A) = B)

Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2860  ℩cio 4338   class class class wbr 4640   ‘cfv 4782

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-br 4641  df-fv 4796

This theorem is referenced by:  tz6.12  5346  tz6.12c  5348  funbrfv  5357  fvfullfunlem3  5864