Theorem sbthlem2 6205

Description: Lemma for sbth 6207. Eliminate hypotheses from sbthlem1 6204. Theorem XI.1.14 of [Rosser] p. 350. (Contributed by SF, 10-Mar-2015.)

Hypothesis

Ref	Expression
sbthlem2.1	⊢ R ∈ V

Assertion

Ref	Expression
sbthlem2	⊢ (((Fun R ∧ Fun ◡R) ∧ (B ∈ V ∧ B ⊆ dom R ∧ ran R ⊆ B)) → ran R ≈ B)

Proof of Theorem sbthlem2

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq1 3293	. . . . . 6 ⊢ (b = B → (b ⊆ dom R ↔ B ⊆ dom R))
2		sseq2 3294	. . . . . 6 ⊢ (b = B → (ran R ⊆ b ↔ ran R ⊆ B))
3	1, 2	anbi12d 691	. . . . 5 ⊢ (b = B → ((b ⊆ dom R ∧ ran R ⊆ b) ↔ (B ⊆ dom R ∧ ran R ⊆ B)))
4		breq2 4644	. . . . . 6 ⊢ (b = B → (ran R ≈ b ↔ ran R ≈ B))
5	4	imbi2d 307	. . . . 5 ⊢ (b = B → (((Fun R ∧ Fun ◡R) → ran R ≈ b) ↔ ((Fun R ∧ Fun ◡R) → ran R ≈ B)))
6	3, 5	imbi12d 311	. . . 4 ⊢ (b = B → (((b ⊆ dom R ∧ ran R ⊆ b) → ((Fun R ∧ Fun ◡R) → ran R ≈ b)) ↔ ((B ⊆ dom R ∧ ran R ⊆ B) → ((Fun R ∧ Fun ◡R) → ran R ≈ B))))
7		sbthlem2.1	. . . . . 6 ⊢ R ∈ V
8		vex 2863	. . . . . 6 ⊢ b ∈ V
9		eqid 2353	. . . . . 6 ⊢ Clos1 ((b ∖ ran R), R) = Clos1 ((b ∖ ran R), R)
10		eqid 2353	. . . . . 6 ⊢ (b ∩ Clos1 ((b ∖ ran R), R)) = (b ∩ Clos1 ((b ∖ ran R), R))
11		eqid 2353	. . . . . 6 ⊢ (b ∖ Clos1 ((b ∖ ran R), R)) = (b ∖ Clos1 ((b ∖ ran R), R))
12		eqid 2353	. . . . . 6 ⊢ (ran R ∩ Clos1 ((b ∖ ran R), R)) = (ran R ∩ Clos1 ((b ∖ ran R), R))
13		eqid 2353	. . . . . 6 ⊢ (ran R ∖ Clos1 ((b ∖ ran R), R)) = (ran R ∖ Clos1 ((b ∖ ran R), R))
14	7, 8, 9, 10, 11, 12, 13	sbthlem1 6204	. . . . 5 ⊢ (((Fun R ∧ Fun ◡R) ∧ (b ⊆ dom R ∧ ran R ⊆ b)) → ran R ≈ b)
15	14	expcom 424	. . . 4 ⊢ ((b ⊆ dom R ∧ ran R ⊆ b) → ((Fun R ∧ Fun ◡R) → ran R ≈ b))
16	6, 15	vtoclg 2915	. . 3 ⊢ (B ∈ V → ((B ⊆ dom R ∧ ran R ⊆ B) → ((Fun R ∧ Fun ◡R) → ran R ≈ B)))
17	16	3impib 1149	. 2 ⊢ ((B ∈ V ∧ B ⊆ dom R ∧ ran R ⊆ B) → ((Fun R ∧ Fun ◡R) → ran R ≈ B))
18	17	impcom 419	1 ⊢ (((Fun R ∧ Fun ◡R) ∧ (B ∈ V ∧ B ⊆ dom R ∧ ran R ⊆ B)) → ran R ≈ B)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ∖ cdif 3207   ∩ cin 3209   ⊆ wss 3258   class class class wbr 4640  ^◡ccnv 4772  dom cdm 4773  ran crn 4774  Fun wfun 4776   Clos1 cclos1 5873   ≈ cen 6029

This proof depends on 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 proof 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-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-2nd 4798  df-txp 5737  df-fix 5741  df-ins2 5751  df-ins3 5753  df-image 5755  df-clos1 5874  df-en 6030

This theorem is used by:  sbthlem3  6206