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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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