Theorem brtcfn 6247

Description: Binary relationship form of the stratified T-raising function. (Contributed by SF, 18-Mar-2015.)

Hypothesis

Ref	Expression
brtcfn.1	⊢ A ∈ V

Assertion

Ref	Expression
brtcfn	⊢ ({A}TcFnB ↔ B = Tc A)

Proof of Theorem brtcfn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brtcfn.1	. . . . 5 ⊢ A ∈ V
2	1	snel1c 4141	. . . 4 ⊢ {A} ∈ 1c
3		unieq 3901	. . . . . . 7 ⊢ (x = {A} → ∪x = ∪{A})
4	1	unisn 3908	. . . . . . 7 ⊢ ∪{A} = A
5	3, 4	syl6eq 2401	. . . . . 6 ⊢ (x = {A} → ∪x = A)
6		tceq 6159	. . . . . 6 ⊢ (∪x = A → Tc ∪x = Tc A)
7	5, 6	syl 15	. . . . 5 ⊢ (x = {A} → Tc ∪x = Tc A)
8		df-tcfn 6108	. . . . 5 ⊢ TcFn = (x ∈ 1c ↦ Tc ∪x)
9		tcex 6158	. . . . 5 ⊢ Tc A ∈ V
10	7, 8, 9	fvmpt 5701	. . . 4 ⊢ ({A} ∈ 1c → (TcFn ‘{A}) = Tc A)
11	2, 10	ax-mp 5	. . 3 ⊢ (TcFn ‘{A}) = Tc A
12	11	eqeq1i 2360	. 2 ⊢ ((TcFn ‘{A}) = B ↔ Tc A = B)
13		fntcfn 6246	. . 3 ⊢ TcFn Fn 1c
14		fnbrfvb 5359	. . 3 ⊢ ((TcFn Fn 1c ∧ {A} ∈ 1c) → ((TcFn ‘{A}) = B ↔ {A}TcFnB))
15	13, 2, 14	mp2an 653	. 2 ⊢ ((TcFn ‘{A}) = B ↔ {A}TcFnB)
16		eqcom 2355	. 2 ⊢ ( Tc A = B ↔ B = Tc A)
17	12, 15, 16	3bitr3i 266	1 ⊢ ({A}TcFnB ↔ B = Tc A)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 2860  {csn 3738  ∪cuni 3892  1_cc1c 4135   class class class wbr 4640   Fn wfn 4777   ‘cfv 4782   T_c ctc 6094  TcFnctcfn 6098

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-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fv 4796  df-mpt 5653  df-tc 6104  df-tcfn 6108

This theorem is referenced by:  nmembers1lem1  6269  nchoicelem11  6300  nchoicelem16  6305