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Theorem sbthlemi6 6955
Description: Lemma for isbth 6960. (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 5033 . . . . 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 4851 . . . . 5 ran 𝐻 = ran ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
8 df-ima 4636 . . . . . 6 (𝑓 𝐷) = ran (𝑓 𝐷)
98uneq1i 3285 . . . . 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 6953 . . . . . 6 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
14 df-ima 4636 . . . . . 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 3289 . . . 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 4977 . . . . . . 7 (𝑓 𝐷) ⊆ ran 𝑓
20 sstr2 3162 . . . . . . 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 6916 . . . . . . 7 (𝐵 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) ↔ ((𝑓 𝐷) ⊆ 𝐵 ∧ ∀𝑦𝐵 DECID 𝑦 ∈ (𝑓 𝐷)))
24 exmidexmid 4193 . . . . . . . . 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 2737  cdif 3126  cun 3127  wss 3129   cuni 3807  EXMIDwem 4191  ccnv 4622  dom cdm 4623  ran crn 4624  cres 4625  cima 4626  Fun wfun 5206
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 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206
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 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-exmid 4192  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-fun 5214
This theorem is referenced by:  sbthlemi9  6958
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