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Theorem sbthlemi5 6957
Description: Lemma for isbth 6963. (Contributed by NM, 22-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
sbthlemi5 ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴)) → dom 𝐻 = 𝐴)
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 sbthlemi5
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 sbthlem.3 . . . . 5 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
21dmeqi 4827 . . . 4 dom 𝐻 = dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
3 dmun 4833 . . . 4 dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (dom (𝑓 𝐷) ∪ dom (𝑔 ↾ (𝐴 𝐷)))
4 dmres 4927 . . . . 5 dom (𝑓 𝐷) = ( 𝐷 ∩ dom 𝑓)
5 dmres 4927 . . . . . 6 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ dom 𝑔)
6 df-rn 4636 . . . . . . . 8 ran 𝑔 = dom 𝑔
76eqcomi 2181 . . . . . . 7 dom 𝑔 = ran 𝑔
87ineq2i 3333 . . . . . 6 ((𝐴 𝐷) ∩ dom 𝑔) = ((𝐴 𝐷) ∩ ran 𝑔)
95, 8eqtri 2198 . . . . 5 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ ran 𝑔)
104, 9uneq12i 3287 . . . 4 (dom (𝑓 𝐷) ∪ dom (𝑔 ↾ (𝐴 𝐷))) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔))
112, 3, 103eqtri 2202 . . 3 dom 𝐻 = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔))
12 sbthlem.1 . . . . . . . . . 10 𝐴 ∈ V
13 sbthlem.2 . . . . . . . . . 10 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
1412, 13sbthlem1 6953 . . . . . . . . 9 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
15 difss 3261 . . . . . . . . 9 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴
1614, 15sstri 3164 . . . . . . . 8 𝐷𝐴
17 sseq2 3179 . . . . . . . 8 (dom 𝑓 = 𝐴 → ( 𝐷 ⊆ dom 𝑓 𝐷𝐴))
1816, 17mpbiri 168 . . . . . . 7 (dom 𝑓 = 𝐴 𝐷 ⊆ dom 𝑓)
19 dfss 3143 . . . . . . 7 ( 𝐷 ⊆ dom 𝑓 𝐷 = ( 𝐷 ∩ dom 𝑓))
2018, 19sylib 122 . . . . . 6 (dom 𝑓 = 𝐴 𝐷 = ( 𝐷 ∩ dom 𝑓))
2120uneq1d 3288 . . . . 5 (dom 𝑓 = 𝐴 → ( 𝐷 ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ (𝐴 𝐷)))
2212, 13sbthlemi3 6955 . . . . . . . 8 ((EXMID ∧ ran 𝑔𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
23 imassrn 4980 . . . . . . . 8 (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔
2422, 23eqsstrrdi 3208 . . . . . . 7 ((EXMID ∧ ran 𝑔𝐴) → (𝐴 𝐷) ⊆ ran 𝑔)
25 dfss 3143 . . . . . . 7 ((𝐴 𝐷) ⊆ ran 𝑔 ↔ (𝐴 𝐷) = ((𝐴 𝐷) ∩ ran 𝑔))
2624, 25sylib 122 . . . . . 6 ((EXMID ∧ ran 𝑔𝐴) → (𝐴 𝐷) = ((𝐴 𝐷) ∩ ran 𝑔))
2726uneq2d 3289 . . . . 5 ((EXMID ∧ ran 𝑔𝐴) → (( 𝐷 ∩ dom 𝑓) ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔)))
2821, 27sylan9eq 2230 . . . 4 ((dom 𝑓 = 𝐴 ∧ (EXMID ∧ ran 𝑔𝐴)) → ( 𝐷 ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔)))
2928an12s 565 . . 3 ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴)) → ( 𝐷 ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔)))
3011, 29eqtr4id 2229 . 2 ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴)) → dom 𝐻 = ( 𝐷 ∪ (𝐴 𝐷)))
31 undifdcss 6919 . . . . 5 (𝐴 = ( 𝐷 ∪ (𝐴 𝐷)) ↔ ( 𝐷𝐴 ∧ ∀𝑦𝐴 DECID 𝑦 𝐷))
32 exmidexmid 4195 . . . . . . 7 (EXMIDDECID 𝑦 𝐷)
3332ralrimivw 2551 . . . . . 6 (EXMID → ∀𝑦𝐴 DECID 𝑦 𝐷)
3433biantrud 304 . . . . 5 (EXMID → ( 𝐷𝐴 ↔ ( 𝐷𝐴 ∧ ∀𝑦𝐴 DECID 𝑦 𝐷)))
3531, 34bitr4id 199 . . . 4 (EXMID → (𝐴 = ( 𝐷 ∪ (𝐴 𝐷)) ↔ 𝐷𝐴))
3616, 35mpbiri 168 . . 3 (EXMID𝐴 = ( 𝐷 ∪ (𝐴 𝐷)))
3736adantr 276 . 2 ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴)) → 𝐴 = ( 𝐷 ∪ (𝐴 𝐷)))
3830, 37eqtr4d 2213 1 ((EXMID ∧ (dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴)) → dom 𝐻 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  DECID wdc 834   = wceq 1353  wcel 2148  {cab 2163  wral 2455  Vcvv 2737  cdif 3126  cun 3127  cin 3128  wss 3129   cuni 3809  EXMIDwem 4193  ccnv 4624  dom cdm 4625  ran crn 4626  cres 4627  cima 4628
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 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-exmid 4194  df-xp 4631  df-cnv 4633  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638
This theorem is referenced by:  sbthlemi9  6961
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