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Theorem sbthlemi8 6818
Description: Lemma for isbth 6821. (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
sbthlemi8 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
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 sbthlemi8
StepHypRef Expression
1 funres11 5163 . . . 4 (Fun 𝑓 → Fun (𝑓 𝐷))
21ad2antlr 478 . . 3 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun (𝑓 𝐷))
3 funcnvcnv 5150 . . . . . 6 (Fun 𝑔 → Fun 𝑔)
4 funres11 5163 . . . . . 6 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
53, 4syl 14 . . . . 5 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
65ad2antrr 477 . . . 4 (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) → Fun (𝑔 ↾ (𝐴 𝐷)))
76ad2antrl 479 . . 3 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun (𝑔 ↾ (𝐴 𝐷)))
8 simpll 501 . . . 4 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → EXMID)
9 simprll 509 . . . . 5 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (Fun 𝑔 ∧ dom 𝑔 = 𝐵))
109simprd 113 . . . 4 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → dom 𝑔 = 𝐵)
11 simprlr 510 . . . 4 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝑔𝐴)
12 simprr 504 . . . 4 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝑔)
13 df-ima 4520 . . . . . . 7 (𝑓 𝐷) = ran (𝑓 𝐷)
14 df-rn 4518 . . . . . . 7 ran (𝑓 𝐷) = dom (𝑓 𝐷)
1513, 14eqtr2i 2137 . . . . . 6 dom (𝑓 𝐷) = (𝑓 𝐷)
16 df-ima 4520 . . . . . . . 8 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
17 df-rn 4518 . . . . . . . 8 ran (𝑔 ↾ (𝐴 𝐷)) = dom (𝑔 ↾ (𝐴 𝐷))
1816, 17eqtri 2136 . . . . . . 7 (𝑔 “ (𝐴 𝐷)) = dom (𝑔 ↾ (𝐴 𝐷))
19 sbthlem.1 . . . . . . . 8 𝐴 ∈ V
20 sbthlem.2 . . . . . . . 8 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
2119, 20sbthlemi4 6814 . . . . . . 7 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
2218, 21syl5eqr 2162 . . . . . 6 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → dom (𝑔 ↾ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
23 ineq12 3240 . . . . . 6 ((dom (𝑓 𝐷) = (𝑓 𝐷) ∧ dom (𝑔 ↾ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷))) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ((𝑓 𝐷) ∩ (𝐵 ∖ (𝑓 𝐷))))
2415, 22, 23sylancr 408 . . . . 5 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ((𝑓 𝐷) ∩ (𝐵 ∖ (𝑓 𝐷))))
25 disjdif 3403 . . . . 5 ((𝑓 𝐷) ∩ (𝐵 ∖ (𝑓 𝐷))) = ∅
2624, 25syl6eq 2164 . . . 4 ((EXMID ∧ (dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
278, 10, 11, 12, 26syl121anc 1204 . . 3 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
28 funun 5135 . . 3 (((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) ∧ (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
292, 7, 27, 28syl21anc 1198 . 2 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
30 sbthlem.3 . . . . 5 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
3130cnveqi 4682 . . . 4 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
32 cnvun 4912 . . . 4 ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
3331, 32eqtri 2136 . . 3 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
3433funeqi 5112 . 2 (Fun 𝐻 ↔ Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
3529, 34sylibr 133 1 (((EXMID ∧ Fun 𝑓) ∧ (((Fun 𝑔 ∧ dom 𝑔 = 𝐵) ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → Fun 𝐻)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 945   = wceq 1314  wcel 1463  {cab 2101  Vcvv 2658  cdif 3036  cun 3037  cin 3038  wss 3039  c0 3331   cuni 3704  EXMIDwem 4086  ccnv 4506  dom cdm 4507  ran crn 4508  cres 4509  cima 4510  Fun wfun 5085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-stab 799  df-dc 803  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-exmid 4087  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-fun 5093
This theorem is referenced by:  sbthlemi9  6819
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