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Theorem sbthlem5 9089
Description: Lemma for sbth 9095. (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 5903 . . . 4 dom 𝐻 = dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
3 dmun 5909 . . . . 5 dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (dom (𝑓 𝐷) ∪ dom (𝑔 ↾ (𝐴 𝐷)))
4 dmres 6002 . . . . . 6 dom (𝑓 𝐷) = ( 𝐷 ∩ dom 𝑓)
5 dmres 6002 . . . . . . 7 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ dom 𝑔)
6 df-rn 5686 . . . . . . . . 9 ran 𝑔 = dom 𝑔
76eqcomi 2739 . . . . . . . 8 dom 𝑔 = ran 𝑔
87ineq2i 4208 . . . . . . 7 ((𝐴 𝐷) ∩ dom 𝑔) = ((𝐴 𝐷) ∩ ran 𝑔)
95, 8eqtri 2758 . . . . . 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 2758 . . . 4 dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔))
122, 11eqtri 2758 . . 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 9085 . . . . . . . 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 9087 . . . . . . 7 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
24 imassrn 6069 . . . . . . 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 2790 . . 3 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → ( 𝐷 ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔)))
3012, 29eqtr4id 2789 . 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 2786 1 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → dom 𝐻 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  {cab 2707  Vcvv 3472  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 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  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  9093
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