Theorem orvcval4 32727

Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 32724. (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.3	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑆MblFnM(sigaGen‘𝐽)))
2	1	isanmbfm 32523	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ∪ ran MblFnM)
3	2	mbfmfun 32519	. . 3 ⊢ (𝜑 → Fun 𝑋)
4		orvccel.1	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
5		orvccel.2	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
6	5	sgsiga 32408	. . . . . 6 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
7	4, 6, 1	mbfmf 32520	. . . . 5 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽))
8		elex 3459	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V)
9		unisg 32409	. . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
10	5, 8, 9	3syl 18	. . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
11	10	feq3d 6638	. . . . 5 ⊢ (𝜑 → (𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽) ↔ 𝑋:∪ 𝑆⟶∪ 𝐽))
12	7, 11	mpbid 231	. . . 4 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ 𝐽)
13	12	frnd 6659	. . 3 ⊢ (𝜑 → ran 𝑋 ⊆ ∪ 𝐽)
14		fimacnvinrn2 7006	. . 3 ⊢ ((Fun 𝑋 ∧ ran 𝑋 ⊆ ∪ 𝐽) → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)))
15	3, 13, 14	syl2anc 584	. 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)))
16		orvccel.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17	3, 1, 16	orvcval 32724	. 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
18		dfrab2 4257	. . . 4 ⊢ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)
19	18	a1i 11	. . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽))
20	19	imaeq2d 5999	. 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)))
21	15, 17, 20	3eqtr4d 2786	1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  {cab 2713  {crab 3403  Vcvv 3441   ∩ cin 3897   ⊆ wss 3898  ∪ cuni 4852   class class class wbr 5092  ^◡ccnv 5619  ran crn 5621   “ cima 5623  Fun wfun 6473  ⟶wf 6475  ‘cfv 6479  (class class class)co 7337  Topctop 22148  sigAlgebracsiga 32374  sigaGencsigagen 32404  MblFnMcmbfm 32515  ∘_RV/𝑐corvc 32722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fo 6485  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-map 8688  df-siga 32375  df-sigagen 32405  df-mbfm 32516  df-orvc 32723

This theorem is referenced by:  orvcoel  32728  orvccel  32729  orrvcval4  32731