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Theorem sbthlemi6 6963
Description: Lemma for isbth 6968. (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 5039 . . . . 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 4857 . . . . 5 ran 𝐻 = ran ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
8 df-ima 4641 . . . . . 6 (𝑓 𝐷) = ran (𝑓 𝐷)
98uneq1i 3287 . . . . 5 ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷))) = (ran (𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷)))
105, 7, 93eqtr4i 2208 . . . 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 6961 . . . . . 6 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
14 df-ima 4641 . . . . . 6 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
1513, 14eqtr3di 2225 . . . . 5 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷)))
1615uneq2d 3291 . . . 4 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷))))
1710, 16eqtr4id 2229 . . 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 4983 . . . . . . 7 (𝑓 𝐷) ⊆ ran 𝑓
20 sstr2 3164 . . . . . . 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 6924 . . . . . . 7 (𝐵 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) ↔ ((𝑓 𝐷) ⊆ 𝐵 ∧ ∀𝑦𝐵 DECID 𝑦 ∈ (𝑓 𝐷)))
24 exmidexmid 4198 . . . . . . . . 9 (EXMIDDECID 𝑦 ∈ (𝑓 𝐷))
2524ralrimivw 2551 . . . . . . . 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 2183 . . 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 2210 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 2148  {cab 2163  wral 2455  Vcvv 2739  cdif 3128  cun 3129  wss 3131   cuni 3811  EXMIDwem 4196  ccnv 4627  dom cdm 4628  ran crn 4629  cres 4630  cima 4631  Fun wfun 5212
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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-exmid 4197  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-fun 5220
This theorem is referenced by:  sbthlemi9  6966
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