Theorem sbthlem2 6204

Description: Lemma for sbth 6206. Eliminate hypotheses from sbthlem1 6203. 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 3292	. . . . . 6 ⊢ (b = B → (b ⊆ dom R ↔ B ⊆ dom R))
2		sseq2 3293	. . . . . 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 4643	. . . . . 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 2862	. . . . . 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 6203	. . . . 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 2914	. . 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)

Syntax hints:   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Vcvv 2859   ∖ cdif 3206   ∩ cin 3208   ⊆ wss 3257   class class class wbr 4639  ^◡ccnv 4771  dom cdm 4772  ran crn 4773  Fun wfun 4775   Clos1 cclos1 5872   ≈ cen 6028

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 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-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-2nd 4797  df-txp 5736  df-fix 5740  df-ins2 5750  df-ins3 5752  df-image 5754  df-clos1 5873  df-en 6029

This theorem is referenced by:  sbthlem3  6205