New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  sbthlem2 GIF version

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 RB)

Proof of Theorem sbthlem2
Dummy variable b is distinct from all other variables.
StepHypRef Expression
1 sseq1 3292 . . . . . 6 (b = B → (b dom RB dom R))
2 sseq2 3293 . . . . . 6 (b = B → (ran R b ↔ ran R B))
31, 2anbi12d 691 . . . . 5 (b = B → ((b dom R ran R b) ↔ (B dom R ran R B)))
4 breq2 4643 . . . . . 6 (b = B → (ran Rb ↔ ran RB))
54imbi2d 307 . . . . 5 (b = B → (((Fun R Fun R) → ran Rb) ↔ ((Fun R Fun R) → ran RB)))
63, 5imbi12d 311 . . . 4 (b = B → (((b dom R ran R b) → ((Fun R Fun R) → ran Rb)) ↔ ((B dom R ran R B) → ((Fun R Fun R) → ran RB))))
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))
147, 8, 9, 10, 11, 12, 13sbthlem1 6203 . . . . 5 (((Fun R Fun R) (b dom R ran R b)) → ran Rb)
1514expcom 424 . . . 4 ((b dom R ran R b) → ((Fun R Fun R) → ran Rb))
166, 15vtoclg 2914 . . 3 (B V → ((B dom R ran R B) → ((Fun R Fun R) → ran RB)))
17163impib 1149 . 2 ((B V B dom R ran R B) → ((Fun R Fun R) → ran RB))
1817impcom 419 1 (((Fun R Fun R) (B V B dom R ran R B)) → ran RB)
 Colors of variables: wff setvar class 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-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-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
 Copyright terms: Public domain W3C validator