Theorem orvcval4 29849

Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 29846. (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 29532	. . . . 5 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
4		orvccel.3	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
5	1, 3, 4	isanmbfm 29645	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ∪ ran MblFnM)
6	5	mbfmfun 29643	. . 3 ⊢ (𝜑 → Fun 𝑋)
7	1, 3, 4	mbfmf 29644	. . . . 5 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽))
8		elex 3185	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V)
9		unisg 29533	. . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
10	2, 8, 9	3syl 18	. . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
11	10	feq3d 5945	. . . . 5 ⊢ (𝜑 → (𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽) ↔ 𝑋:∪ 𝑆⟶∪ 𝐽))
12	7, 11	mpbid 221	. . . 4 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ 𝐽)
13		frn 5966	. . . 4 ⊢ (𝑋:∪ 𝑆⟶∪ 𝐽 → ran 𝑋 ⊆ ∪ 𝐽)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → ran 𝑋 ⊆ ∪ 𝐽)
15		fimacnvinrn2 6257	. . 3 ⊢ ((Fun 𝑋 ∧ ran 𝑋 ⊆ ∪ 𝐽) → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)))
16	6, 14, 15	syl2anc 691	. 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)))
17		orvccel.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
18	6, 4, 17	orvcval 29846	. 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
19		dfrab2 3862	. . . 4 ⊢ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)
20	19	a1i 11	. . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽))
21	20	imaeq2d 5385	. 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)))
22	16, 18, 21	3eqtr4d 2654	1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {cab 2596  {crab 2900  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583  ^◡ccnv 5037  ran crn 5039   “ cima 5041  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Topctop 20517  sigAlgebracsiga 29497  sigaGencsigagen 29528  MblFnMcmbfm 29639  ∘_RV/𝑐corvc 29844

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-siga 29498  df-sigagen 29529  df-mbfm 29640  df-orvc 29845

This theorem is referenced by:  orvcoel  29850  orvccel  29851  orrvcval4  29853