Theorem orvcval4 32427

Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 32424. (Contributed by Thierry Arnoux, 21-Jan-2017.)

Hypotheses

Ref	Expression
orvccel.1	⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
orvccel.2	⊢ (𝜑 → 𝐽 ∈ Top)
orvccel.3	⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
orvccel.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)

Assertion

Ref	Expression
orvcval4	⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}))

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅   𝑦,𝑋   𝑦,𝐽

Allowed substitution hints:   𝜑(𝑦)   𝑆(𝑦)   𝑉(𝑦)

Proof of Theorem orvcval4

Step	Hyp	Ref	Expression
1		orvccel.1	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
2		orvccel.2	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Top)
3	2	sgsiga 32110	. . . . 5 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
4		orvccel.3	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
5	1, 3, 4	isanmbfm 32223	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ∪ ran MblFnM)
6	5	mbfmfun 32221	. . 3 ⊢ (𝜑 → Fun 𝑋)
7	1, 3, 4	mbfmf 32222	. . . . 5 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽))
8		elex 3450	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V)
9		unisg 32111	. . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
10	2, 8, 9	3syl 18	. . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
11	10	feq3d 6587	. . . . 5 ⊢ (𝜑 → (𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽) ↔ 𝑋:∪ 𝑆⟶∪ 𝐽))
12	7, 11	mpbid 231	. . . 4 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ 𝐽)
13	12	frnd 6608	. . 3 ⊢ (𝜑 → ran 𝑋 ⊆ ∪ 𝐽)
14		fimacnvinrn2 6950	. . 3 ⊢ ((Fun 𝑋 ∧ ran 𝑋 ⊆ ∪ 𝐽) → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)))
15	6, 13, 14	syl2anc 584	. 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)))
16		orvccel.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17	6, 4, 16	orvcval 32424	. 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
18		dfrab2 4244	. . . 4 ⊢ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)
19	18	a1i 11	. . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽))
20	19	imaeq2d 5969	. 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)))
21	15, 17, 20	3eqtr4d 2788	1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  {cab 2715  {crab 3068  Vcvv 3432   ∩ cin 3886   ⊆ wss 3887  ∪ cuni 4839   class class class wbr 5074  ^◡ccnv 5588  ran crn 5590   “ cima 5592  Fun wfun 6427  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  Topctop 22042  sigAlgebracsiga 32076  sigaGencsigagen 32106  MblFnMcmbfm 32217  ∘_RV/𝑐corvc 32422

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-siga 32077  df-sigagen 32107  df-mbfm 32218  df-orvc 32423

This theorem is referenced by:  orvcoel  32428  orvccel  32429  orrvcval4  32431