Theorem orvcval4 34459

Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 34456. (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 34254	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ∪ ran MblFnM)
3	2	mbfmfun 34250	. . 3 ⊢ (𝜑 → Fun 𝑋)
4		orvccel.1	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ∪ ran sigAlgebra)
5		orvccel.2	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top)
6	5	sgsiga 34139	. . . . . 6 ⊢ (𝜑 → (sigaGen‘𝐽) ∈ ∪ ran sigAlgebra)
7	4, 6, 1	mbfmf 34251	. . . . 5 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽))
8		elex 3471	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝐽 ∈ V)
9		unisg 34140	. . . . . . 7 ⊢ (𝐽 ∈ V → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
10	5, 8, 9	3syl 18	. . . . . 6 ⊢ (𝜑 → ∪ (sigaGen‘𝐽) = ∪ 𝐽)
11	10	feq3d 6676	. . . . 5 ⊢ (𝜑 → (𝑋:∪ 𝑆⟶∪ (sigaGen‘𝐽) ↔ 𝑋:∪ 𝑆⟶∪ 𝐽))
12	7, 11	mpbid 232	. . . 4 ⊢ (𝜑 → 𝑋:∪ 𝑆⟶∪ 𝐽)
13	12	frnd 6699	. . 3 ⊢ (𝜑 → ran 𝑋 ⊆ ∪ 𝐽)
14		fimacnvinrn2 7047	. . 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 34456	. 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
18		dfrab2 4286	. . . 4 ⊢ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)
19	18	a1i 11	. . 3 ⊢ (𝜑 → {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴} = ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽))
20	19	imaeq2d 6034	. 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ ({𝑦 ∣ 𝑦𝑅𝐴} ∩ ∪ 𝐽)))
21	15, 17, 20	3eqtr4d 2775	1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ 𝐽 ∣ 𝑦𝑅𝐴}))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {cab 2708  {crab 3408  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917  ∪ cuni 4874   class class class wbr 5110  ^◡ccnv 5640  ran crn 5642   “ cima 5644  Fun wfun 6508  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Topctop 22787  sigAlgebracsiga 34105  sigaGencsigagen 34135  MblFnMcmbfm 34246  ∘_RV/𝑐corvc 34454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-siga 34106  df-sigagen 34136  df-mbfm 34247  df-orvc 34455

This theorem is referenced by:  orvcoel  34460  orvccel  34461  orrvcval4  34463