MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbthlem6 Structured version   Visualization version   GIF version

Theorem sbthlem6 9127
Description: Lemma for sbth 9132. (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 6168 . . . 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 5951 . . . 4 ran 𝐻 = ran ((𝑓 𝐷) ∪ (𝑔 ↾ (𝐴 𝐷)))
4 df-ima 5702 . . . . 5 (𝑓 𝐷) = ran (𝑓 𝐷)
54uneq1i 4174 . . . 4 ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷))) = (ran (𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷)))
61, 3, 53eqtr4i 2773 . . 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 9125 . . . . 5 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝑔 “ (𝐴 𝐷)) = (𝐵 ∖ (𝑓 𝐷)))
10 df-ima 5702 . . . . 5 (𝑔 “ (𝐴 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷))
119, 10eqtr3di 2790 . . . 4 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → (𝐵 ∖ (𝑓 𝐷)) = ran (𝑔 ↾ (𝐴 𝐷)))
1211uneq2d 4178 . . 3 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = ((𝑓 𝐷) ∪ ran (𝑔 ↾ (𝐴 𝐷))))
136, 12eqtr4id 2794 . 2 (((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔) → ran 𝐻 = ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))))
14 imassrn 6091 . . . 4 (𝑓 𝐷) ⊆ ran 𝑓
15 sstr2 4002 . . . 4 ((𝑓 𝐷) ⊆ ran 𝑓 → (ran 𝑓𝐵 → (𝑓 𝐷) ⊆ 𝐵))
1614, 15ax-mp 5 . . 3 (ran 𝑓𝐵 → (𝑓 𝐷) ⊆ 𝐵)
17 undif 4488 . . 3 ((𝑓 𝐷) ⊆ 𝐵 ↔ ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = 𝐵)
1816, 17sylib 218 . 2 (ran 𝑓𝐵 → ((𝑓 𝐷) ∪ (𝐵 ∖ (𝑓 𝐷))) = 𝐵)
1913, 18sylan9eqr 2797 1 ((ran 𝑓𝐵 ∧ ((dom 𝑔 = 𝐵 ∧ ran 𝑔𝐴) ∧ Fun 𝑔)) → ran 𝐻 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {cab 2712  Vcvv 3478  cdif 3960  cun 3961  wss 3963   cuni 4912  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  cima 5692  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565
This theorem is referenced by:  sbthlem9  9130
  Copyright terms: Public domain W3C validator