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Theorem sbthlem6 9121
Description: Lemma for sbth 9126. (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
sbthlem6 ((ran 𝑓𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
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 sbthlem6
StepHypRef Expression
1 rnun 6155 . . . 4 ran ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷))) = (ran (𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷)))
2 sbthlem.3 . . . . 5 𝐻 = ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
32rneqi 5943 . . . 4 ran 𝐻 = ran ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
4 df-ima 5695 . . . . 5 (𝑓 𝐷) = ran (𝑓 𝐷)
54uneq1i 4160 . . . 4 ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷))) = (ran (𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷)))
61, 3, 53eqtr4i 2766 . . 3 ran 𝐻 = ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷)))
7 sbthlem.1 . . . . . 6 𝐴 ∈ V
8 sbthlem.2 . . . . . 6 𝐷 = {𝑥 ∣ (𝑥𝐴 ∧ (𝑔 “ (𝐵 ∖ (𝑓𝑥))) ⊆ (𝐴𝑥))}
97, 8sbthlem4 9119 . . . . 5 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
10 df-ima 5695 . . . . 5 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
119, 10eqtr3di 2783 . . . 4 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷)))
1211uneq2d 4164 . . 3 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷))))
136, 12eqtr4id 2787 . 2 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → ran 𝐻 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))))
14 imassrn 6079 . . . 4 (𝑓 𝐷) ⊆ ran 𝑓
15 sstr2 3989 . . . 4 ((𝑓 𝐷) ⊆ ran 𝑓 → (ran 𝑓𝐵 → (𝑓 𝐷) ⊆ 𝐵))
1614, 15ax-mp 5 . . 3 (ran 𝑓𝐵 → (𝑓 𝐷) ⊆ 𝐵)
17 undif 4485 . . 3 ((𝑓 𝐷) ⊆ 𝐵 ↔ ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = 𝐵)
1816, 17sylib 217 . 2 (ran 𝑓𝐵 → ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = 𝐵)
1913, 18sylan9eqr 2790 1 ((ran 𝑓𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {cab 2705  Vcvv 3473  cdif 3946  cun 3947  wss 3949   cuni 4912  ccnv 5681  dom cdm 5682  ran crn 5683  cres 5684  cima 5685  Fun wfun 6547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-fun 6555
This theorem is referenced by:  sbthlem9  9124
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