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Theorem sbthlem5 9038
Description: Lemma for sbth 9044. (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 5865 . . . 4 dom 𝐻 = dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
3 dmun 5871 . . . . 5 dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (dom (𝑓 𝐷) ∪ dom (𝑔 ↾ (𝐴 𝐷)))
4 dmres 5964 . . . . . 6 dom (𝑓 𝐷) = ( 𝐷 ∩ dom 𝑓)
5 dmres 5964 . . . . . . 7 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ dom 𝑔)
6 df-rn 5649 . . . . . . . . 9 ran 𝑔 = dom 𝑔
76eqcomi 2746 . . . . . . . 8 dom 𝑔 = ran 𝑔
87ineq2i 4174 . . . . . . 7 ((𝐴 𝐷) ∩ dom 𝑔) = ((𝐴 𝐷) ∩ ran 𝑔)
95, 8eqtri 2765 . . . . . 6 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ ran 𝑔)
104, 9uneq12i 4126 . . . . 5 (dom (𝑓 𝐷) ∪ dom (𝑔 ↾ (𝐴 𝐷))) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔))
113, 10eqtri 2765 . . . 4 dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔))
122, 11eqtri 2765 . . 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 9034 . . . . . . . 8 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
16 difss 4096 . . . . . . . 8 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴
1715, 16sstri 3958 . . . . . . 7 𝐷𝐴
18 sseq2 3975 . . . . . . 7 (dom 𝑓 = 𝐴 → ( 𝐷 ⊆ dom 𝑓 𝐷𝐴))
1917, 18mpbiri 258 . . . . . 6 (dom 𝑓 = 𝐴 𝐷 ⊆ dom 𝑓)
20 dfss 3933 . . . . . 6 ( 𝐷 ⊆ dom 𝑓 𝐷 = ( 𝐷 ∩ dom 𝑓))
2119, 20sylib 217 . . . . 5 (dom 𝑓 = 𝐴 𝐷 = ( 𝐷 ∩ dom 𝑓))
2221uneq1d 4127 . . . 4 (dom 𝑓 = 𝐴 → ( 𝐷 ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ (𝐴 𝐷)))
2313, 14sbthlem3 9036 . . . . . . 7 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
24 imassrn 6029 . . . . . . 7 (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔
2523, 24eqsstrrdi 4004 . . . . . 6 (ran 𝑔𝐴 → (𝐴 𝐷) ⊆ ran 𝑔)
26 dfss 3933 . . . . . 6 ((𝐴 𝐷) ⊆ ran 𝑔 ↔ (𝐴 𝐷) = ((𝐴 𝐷) ∩ ran 𝑔))
2725, 26sylib 217 . . . . 5 (ran 𝑔𝐴 → (𝐴 𝐷) = ((𝐴 𝐷) ∩ ran 𝑔))
2827uneq2d 4128 . . . 4 (ran 𝑔𝐴 → (( 𝐷 ∩ dom 𝑓) ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔)))
2922, 28sylan9eq 2797 . . 3 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → ( 𝐷 ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔)))
3012, 29eqtr4id 2796 . 2 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → dom 𝐻 = ( 𝐷 ∪ (𝐴 𝐷)))
31 undif 4446 . . 3 ( 𝐷𝐴 ↔ ( 𝐷 ∪ (𝐴 𝐷)) = 𝐴)
3217, 31mpbi 229 . 2 ( 𝐷 ∪ (𝐴 𝐷)) = 𝐴
3330, 32eqtrdi 2793 1 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → dom 𝐻 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2714  Vcvv 3448  cdif 3912  cun 3913  cin 3914  wss 3915   cuni 4870  ccnv 5637  dom cdm 5638  ran crn 5639  cres 5640  cima 5641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651
This theorem is referenced by:  sbthlem9  9042
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