ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sbthlemi6 GIF version

Theorem sbthlemi6 6951
Description: Lemma for isbth 6956. (Contributed by NM, 27-Mar-1998.)
Hypotheses
Ref Expression
sbthlem.1 𝐴 ∈ V
sbthlem.2 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
sbthlem.3 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
Assertion
Ref Expression
sbthlemi6 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝑓   𝑥,𝑔   𝑥,𝐻
Allowed substitution hints:   𝐴(𝑓,𝑔)   𝐵(𝑓,𝑔)   𝐷(𝑓,𝑔)   𝐻(𝑓,𝑔)

Proof of Theorem sbthlemi6
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 simpll 527 . . 3 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → EXMID)
2 simprll 537 . . 3 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝑔 = 𝐵)
3 simprlr 538 . . 3 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝑔𝐴)
4 simprr 531 . . 3 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝑔)
5 rnun 5029 . . . . 5 ran ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (ran (𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷)))
6 sbthlem.3 . . . . . 6 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
76rneqi 4848 . . . . 5 ran 𝐻 = ran ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
8 df-ima 4633 . . . . . 6 (𝑓 𝐷) = ran (𝑓 𝐷)
98uneq1i 3283 . . . . 5 ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷))) = (ran (𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷)))
105, 7, 93eqtr4i 2206 . . . 4 ran 𝐻 = ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷)))
11 sbthlem.1 . . . . . . 7 𝐴 ∈ V
12 sbthlem.2 . . . . . . 7 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
1311, 12sbthlemi4 6949 . . . . . 6 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
14 df-ima 4633 . . . . . 6 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
1513, 14eqtr3di 2223 . . . . 5 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷)))
1615uneq2d 3287 . . . 4 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷))))
1710, 16eqtr4id 2227 . . 3 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → ran 𝐻 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))))
181, 2, 3, 4, 17syl121anc 1243 . 2 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))))
19 imassrn 4974 . . . . . . 7 (𝑓 𝐷) ⊆ ran 𝑓
20 sstr2 3160 . . . . . . 7 ((𝑓 𝐷) ⊆ ran 𝑓 → (ran 𝑓𝐵 → (𝑓 𝐷) ⊆ 𝐵))
2119, 20ax-mp 5 . . . . . 6 (ran 𝑓𝐵 → (𝑓 𝐷) ⊆ 𝐵)
2221adantl 277 . . . . 5 ((EXMID ∧ ran 𝑓𝐵) → (𝑓 𝐷) ⊆ 𝐵)
23 undifdcss 6912 . . . . . . 7 (𝐵 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) ↔ ((𝑓 𝐷) ⊆ 𝐵 ∧ ∀𝑦𝐵 DECID 𝑦 ∈ (𝑓 𝐷)))
24 exmidexmid 4191 . . . . . . . . 9 (EXMIDDECID 𝑦 ∈ (𝑓 𝐷))
2524ralrimivw 2549 . . . . . . . 8 (EXMID → ∀𝑦𝐵 DECID 𝑦 ∈ (𝑓 𝐷))
2625biantrud 304 . . . . . . 7 (EXMID → ((𝑓 𝐷) ⊆ 𝐵 ↔ ((𝑓 𝐷) ⊆ 𝐵 ∧ ∀𝑦𝐵 DECID 𝑦 ∈ (𝑓 𝐷))))
2723, 26bitr4id 199 . . . . . 6 (EXMID → (𝐵 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) ↔ (𝑓 𝐷) ⊆ 𝐵))
2827adantr 276 . . . . 5 ((EXMID ∧ ran 𝑓𝐵) → (𝐵 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) ↔ (𝑓 𝐷) ⊆ 𝐵))
2922, 28mpbird 167 . . . 4 ((EXMID ∧ ran 𝑓𝐵) → 𝐵 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))))
3029eqcomd 2181 . . 3 ((EXMID ∧ ran 𝑓𝐵) → ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = 𝐵)
3130adantr 276 . 2 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = 𝐵)
3218, 31eqtrd 2208 1 (((EXMID ∧ ran 𝑓𝐵) ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  DECID wdc 834  w3a 978   = wceq 1353  wcel 2146  {cab 2161  wral 2453  Vcvv 2735  cdif 3124  cun 3125  wss 3127   cuni 3805  EXMIDwem 4189  ccnv 4619  dom cdm 4620  ran crn 4621  cres 4622  cima 4623  Fun wfun 5202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-nul 4124  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-exmid 4190  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-fun 5210
This theorem is referenced by:  sbthlemi9  6954
  Copyright terms: Public domain W3C validator