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Theorem sbthlem7 6975
Description: Lemma for isbth 6979. (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
sbthlem7 ((Fun 𝑓 ∧ 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 sbthlem7
StepHypRef Expression
1 funres 5269 . . 3 (Fun 𝑓 → Fun (𝑓 𝐷))
2 funres 5269 . . 3 (Fun 𝑔 → Fun (𝑔 ↾ (𝐴 𝐷)))
3 dmres 4940 . . . . . . . . 9 dom (𝑓 𝐷) = ( 𝐷 ∩ dom 𝑓)
4 inss1 3367 . . . . . . . . 9 ( 𝐷 ∩ dom 𝑓) ⊆ 𝐷
53, 4eqsstri 3199 . . . . . . . 8 dom (𝑓 𝐷) ⊆ 𝐷
6 ssrin 3372 . . . . . . . 8 (dom (𝑓 𝐷) ⊆ 𝐷 → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ dom (𝑔 ↾ (𝐴 𝐷))))
75, 6ax-mp 5 . . . . . . 7 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ dom (𝑔 ↾ (𝐴 𝐷)))
8 dmres 4940 . . . . . . . . 9 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ dom 𝑔)
9 inss1 3367 . . . . . . . . 9 ((𝐴 𝐷) ∩ dom 𝑔) ⊆ (𝐴 𝐷)
108, 9eqsstri 3199 . . . . . . . 8 dom (𝑔 ↾ (𝐴 𝐷)) ⊆ (𝐴 𝐷)
11 sslin 3373 . . . . . . . 8 (dom (𝑔 ↾ (𝐴 𝐷)) ⊆ (𝐴 𝐷) → ( 𝐷 ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ (𝐴 𝐷)))
1210, 11ax-mp 5 . . . . . . 7 ( 𝐷 ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ (𝐴 𝐷))
137, 12sstri 3176 . . . . . 6 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ( 𝐷 ∩ (𝐴 𝐷))
14 disjdif 3507 . . . . . 6 ( 𝐷 ∩ (𝐴 𝐷)) = ∅
1513, 14sseqtri 3201 . . . . 5 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ∅
16 ss0 3475 . . . . 5 ((dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) ⊆ ∅ → (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅)
1715, 16ax-mp 5 . . . 4 (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅
18 funun 5272 . . . 4 (((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) ∧ (dom (𝑓 𝐷) ∩ dom (𝑔 ↾ (𝐴 𝐷))) = ∅) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
1917, 18mpan2 425 . . 3 ((Fun (𝑓 𝐷) ∧ Fun (𝑔 ↾ (𝐴 𝐷))) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
201, 2, 19syl2an 289 . 2 ((Fun 𝑓 ∧ Fun 𝑔) → Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
21 sbthlem.3 . . 3 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
2221funeqi 5249 . 2 (Fun 𝐻 ↔ Fun ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))))
2320, 22sylibr 134 1 ((Fun 𝑓 ∧ Fun 𝑔) → Fun 𝐻)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wcel 2158  {cab 2173  Vcvv 2749  cdif 3138  cun 3139  cin 3140  wss 3141  c0 3434   cuni 3821  ccnv 4637  dom cdm 4638  cres 4640  cima 4641  Fun wfun 5222
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-res 4650  df-fun 5230
This theorem is referenced by:  sbthlemi9  6977
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