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Theorem sbthlem5 8617
 Description: Lemma for sbth 8623. (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.1 . . . . . . . . 9 𝐴 ∈ V
2 sbthlem.2 . . . . . . . . 9 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
31, 2sbthlem1 8613 . . . . . . . 8 𝐷 ⊆ (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷))))
4 difss 4092 . . . . . . . 8 (𝐴 ∖ (𝑔 “ (𝐵 ∖ (𝑓 𝐷)))) ⊆ 𝐴
53, 4sstri 3960 . . . . . . 7 𝐷𝐴
6 sseq2 3977 . . . . . . 7 (dom 𝑓 = 𝐴 → ( 𝐷 ⊆ dom 𝑓 𝐷𝐴))
75, 6mpbiri 261 . . . . . 6 (dom 𝑓 = 𝐴 𝐷 ⊆ dom 𝑓)
8 dfss 3936 . . . . . 6 ( 𝐷 ⊆ dom 𝑓 𝐷 = ( 𝐷 ∩ dom 𝑓))
97, 8sylib 221 . . . . 5 (dom 𝑓 = 𝐴 𝐷 = ( 𝐷 ∩ dom 𝑓))
109uneq1d 4122 . . . 4 (dom 𝑓 = 𝐴 → ( 𝐷 ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ (𝐴 𝐷)))
111, 2sbthlem3 8615 . . . . . . 7 (ran 𝑔𝐴 → (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) = (𝐴 𝐷))
12 imassrn 5923 . . . . . . 7 (𝑔 “ (𝐵 ∖ (𝑓 𝐷))) ⊆ ran 𝑔
1311, 12eqsstrrdi 4006 . . . . . 6 (ran 𝑔𝐴 → (𝐴 𝐷) ⊆ ran 𝑔)
14 dfss 3936 . . . . . 6 ((𝐴 𝐷) ⊆ ran 𝑔 ↔ (𝐴 𝐷) = ((𝐴 𝐷) ∩ ran 𝑔))
1513, 14sylib 221 . . . . 5 (ran 𝑔𝐴 → (𝐴 𝐷) = ((𝐴 𝐷) ∩ ran 𝑔))
1615uneq2d 4123 . . . 4 (ran 𝑔𝐴 → (( 𝐷 ∩ dom 𝑓) ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔)))
1710, 16sylan9eq 2879 . . 3 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → ( 𝐷 ∪ (𝐴 𝐷)) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔)))
18 sbthlem.3 . . . . 5 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
1918dmeqi 5756 . . . 4 dom 𝐻 = dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
20 dmun 5762 . . . . 5 dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (dom (𝑓 𝐷) ∪ dom (𝑔 ↾ (𝐴 𝐷)))
21 dmres 5858 . . . . . 6 dom (𝑓 𝐷) = ( 𝐷 ∩ dom 𝑓)
22 dmres 5858 . . . . . . 7 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ dom 𝑔)
23 df-rn 5549 . . . . . . . . 9 ran 𝑔 = dom 𝑔
2423eqcomi 2833 . . . . . . . 8 dom 𝑔 = ran 𝑔
2524ineq2i 4169 . . . . . . 7 ((𝐴 𝐷) ∩ dom 𝑔) = ((𝐴 𝐷) ∩ ran 𝑔)
2622, 25eqtri 2847 . . . . . 6 dom (𝑔 ↾ (𝐴 𝐷)) = ((𝐴 𝐷) ∩ ran 𝑔)
2721, 26uneq12i 4121 . . . . 5 (dom (𝑓 𝐷) ∪ dom (𝑔 ↾ (𝐴 𝐷))) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔))
2820, 27eqtri 2847 . . . 4 dom ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔))
2919, 28eqtri 2847 . . 3 dom 𝐻 = (( 𝐷 ∩ dom 𝑓) ∪ ((𝐴 𝐷) ∩ ran 𝑔))
3017, 29syl6reqr 2878 . 2 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → dom 𝐻 = ( 𝐷 ∪ (𝐴 𝐷)))
31 undif 4411 . . 3 ( 𝐷𝐴 ↔ ( 𝐷 ∪ (𝐴 𝐷)) = 𝐴)
325, 31mpbi 233 . 2 ( 𝐷 ∪ (𝐴 𝐷)) = 𝐴
3330, 32syl6eq 2875 1 ((dom 𝑓 = 𝐴 ∧ ran 𝑔𝐴) → dom 𝐻 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115  {cab 2802  Vcvv 3479   ∖ cdif 3915   ∪ cun 3916   ∩ cin 3917   ⊆ wss 3918  ∪ cuni 4821  ◡ccnv 5537  dom cdm 5538  ran crn 5539   ↾ cres 5540   “ cima 5541 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-opab 5112  df-xp 5544  df-cnv 5546  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551 This theorem is referenced by:  sbthlem9  8621
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