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Theorem sbthlem5 9083
Description: Lemma for sbth 9089. (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
sbthlem5 ((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 sbthlem5
StepHypRef Expression
1 sbthlem.3 . . . . 5 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
21dmeqi 5902 . . . 4 dom 𝐻 = dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
3 dmun 5908 . . . . 5 dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (dom (𝑓 𝐷) ∪ dom (𝑔 ↾ (𝐴 𝐷)))
4 dmres 6001 . . . . . 6 dom (𝑓 𝐷) = ( 𝐷 ∩ dom 𝑓)
5 dmres 6001 . . . . . . 7 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ dom 𝑔)
6 df-rn 5686 . . . . . . . . 9 ran 𝑔 = dom 𝑔
76eqcomi 2741 . . . . . . . 8 dom 𝑔 = ran 𝑔
87ineq2i 4208 . . . . . . 7 ((𝐴 𝐷) ∩ dom 𝑔) = ((𝐴 𝐷) ∩ ran 𝑔)
95, 8eqtri 2760 . . . . . 6 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ ran 𝑔)
104, 9uneq12i 4160 . . . . 5 (dom (𝑓 𝐷) ∪ dom (𝑔 ↾ (𝐴 𝐷))) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔))
113, 10eqtri 2760 . . . 4 dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔))
122, 11eqtri 2760 . . 3 dom 𝐻 = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔))
13 sbthlem.1 . . . . . . . . 9 𝐴 ∈ V
14 sbthlem.2 . . . . . . . . 9 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
1513, 14sbthlem1 9079 . . . . . . . 8 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
16 difss 4130 . . . . . . . 8 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴
1715, 16sstri 3990 . . . . . . 7 𝐷𝐴
18 sseq2 4007 . . . . . . 7 (dom 𝑓 = 𝐴 → ( 𝐷 ⊆ dom 𝑓 𝐷𝐴))
1917, 18mpbiri 257 . . . . . 6 (dom 𝑓 = 𝐴 𝐷 ⊆ dom 𝑓)
20 dfss 3965 . . . . . 6 ( 𝐷 ⊆ dom 𝑓 𝐷 = ( 𝐷 ∩ dom 𝑓))
2119, 20sylib 217 . . . . 5 (dom 𝑓 = 𝐴 𝐷 = ( 𝐷 ∩ dom 𝑓))
2221uneq1d 4161 . . . 4 (dom 𝑓 = 𝐴 → ( 𝐷 ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ (𝐴 𝐷)))
2313, 14sbthlem3 9081 . . . . . . 7 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
24 imassrn 6068 . . . . . . 7 (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔
2523, 24eqsstrrdi 4036 . . . . . 6 (ran 𝑔𝐴 → (𝐴 𝐷) ⊆ ran 𝑔)
26 dfss 3965 . . . . . 6 ((𝐴 𝐷) ⊆ ran 𝑔 ↔ (𝐴 𝐷) = ((𝐴 𝐷) ∩ ran 𝑔))
2725, 26sylib 217 . . . . 5 (ran 𝑔𝐴 → (𝐴 𝐷) = ((𝐴 𝐷) ∩ ran 𝑔))
2827uneq2d 4162 . . . 4 (ran 𝑔𝐴 → (( 𝐷 ∩ dom 𝑓) ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔)))
2922, 28sylan9eq 2792 . . 3 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → ( 𝐷 ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔)))
3012, 29eqtr4id 2791 . 2 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → dom 𝐻 = ( 𝐷 ∪ (𝐴 𝐷)))
31 undif 4480 . . 3 ( 𝐷𝐴 ↔ ( 𝐷 ∪ (𝐴 𝐷)) = 𝐴)
3217, 31mpbi 229 . 2 ( 𝐷 ∪ (𝐴 𝐷)) = 𝐴
3330, 32eqtrdi 2788 1 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → dom 𝐻 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {cab 2709  Vcvv 3474  cdif 3944  cun 3945  cin 3946  wss 3947   cuni 4907  ccnv 5674  dom cdm 5675  ran crn 5676  cres 5677  cima 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688
This theorem is referenced by:  sbthlem9  9087
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