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Theorem sbthlemi5 6962
Description: Lemma for isbth 6968. (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 4830 . . . 4 dom 𝐻 = dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
3 dmun 4836 . . . 4 dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (dom (𝑓 𝐷) ∪ dom (𝑔 ↾ (𝐴 𝐷)))
4 dmres 4930 . . . . 5 dom (𝑓 𝐷) = ( 𝐷 ∩ dom 𝑓)
5 dmres 4930 . . . . . 6 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ dom 𝑔)
6 df-rn 4639 . . . . . . . 8 ran 𝑔 = dom 𝑔
76eqcomi 2181 . . . . . . 7 dom 𝑔 = ran 𝑔
87ineq2i 3335 . . . . . 6 ((𝐴 𝐷) ∩ dom 𝑔) = ((𝐴 𝐷) ∩ ran 𝑔)
95, 8eqtri 2198 . . . . 5 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ ran 𝑔)
104, 9uneq12i 3289 . . . 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 6958 . . . . . . . . 9 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
15 difss 3263 . . . . . . . . 9 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴
1614, 15sstri 3166 . . . . . . . 8 𝐷𝐴
17 sseq2 3181 . . . . . . . 8 (dom 𝑓 = 𝐴 → ( 𝐷 ⊆ dom 𝑓 𝐷𝐴))
1816, 17mpbiri 168 . . . . . . 7 (dom 𝑓 = 𝐴 𝐷 ⊆ dom 𝑓)
19 dfss 3145 . . . . . . 7 ( 𝐷 ⊆ dom 𝑓 𝐷 = ( 𝐷 ∩ dom 𝑓))
2018, 19sylib 122 . . . . . 6 (dom 𝑓 = 𝐴 𝐷 = ( 𝐷 ∩ dom 𝑓))
2120uneq1d 3290 . . . . 5 (dom 𝑓 = 𝐴 → ( 𝐷 ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ (𝐴 𝐷)))
2212, 13sbthlemi3 6960 . . . . . . . 8 ((EXMID ∧ ran 𝑔𝐴) → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
23 imassrn 4983 . . . . . . . 8 (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔
2422, 23eqsstrrdi 3210 . . . . . . 7 ((EXMID ∧ ran 𝑔𝐴) → (𝐴 𝐷) ⊆ ran 𝑔)
25 dfss 3145 . . . . . . 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 3291 . . . . 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 6924 . . . . 5 (𝐴 = ( 𝐷 ∪ (𝐴 𝐷)) ↔ ( 𝐷𝐴 ∧ ∀𝑦𝐴 DECID 𝑦 𝐷))
32 exmidexmid 4198 . . . . . . 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 2739  cdif 3128  cun 3129  cin 3130  wss 3131   cuni 3811  EXMIDwem 4196  ccnv 4627  dom cdm 4628  ran crn 4629  cres 4630  cima 4631
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-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641
This theorem is referenced by:  sbthlemi9  6966
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