Theorem orvcval2 34790

Description: Another way to express the value of the preimage mapping operator. (Contributed by Thierry Arnoux, 19-Jan-2017.)

Hypotheses

Ref	Expression
orvcval.1	⊢ (𝜑 → Fun 𝑋)
orvcval.2	⊢ (𝜑 → 𝑋 ∈ 𝑉)
orvcval.3	⊢ (𝜑 → 𝐴 ∈ 𝑊)

Assertion

Ref	Expression
orvcval2	⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴})

Distinct variable groups:   𝑧,𝐴   𝑧,𝑅   𝑧,𝑋

Allowed substitution hints:   𝜑(𝑧)   𝑉(𝑧)   𝑊(𝑧)

Proof of Theorem orvcval2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		orvcval.1	. . 3 ⊢ (𝜑 → Fun 𝑋)
2		orvcval.2	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
3		orvcval.3	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊)
4	1, 2, 3	orvcval 34789	. 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}))
5		funfn 6563	. . . 4 ⊢ (Fun 𝑋 ↔ 𝑋 Fn dom 𝑋)
6	1, 5	sylib 221	. . 3 ⊢ (𝜑 → 𝑋 Fn dom 𝑋)
7		fncnvima2 7054	. . 3 ⊢ (𝑋 Fn dom 𝑋 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}})
8	6, 7	syl 18	. 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}})
9		fvex 6892	. . . . 5 ⊢ (𝑋‘𝑧) ∈ V
10		breq1 5113	. . . . 5 ⊢ (𝑦 = (𝑋‘𝑧) → (𝑦𝑅𝐴 ↔ (𝑋‘𝑧)𝑅𝐴))
11	9, 10	elab 3647	. . . 4 ⊢ ((𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴} ↔ (𝑋‘𝑧)𝑅𝐴)
12	11	rabbii 3428	. . 3 ⊢ {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴}
13	12	a1i 11	. 2 ⊢ (𝜑 → {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧) ∈ {𝑦 ∣ 𝑦𝑅𝐴}} = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴})
14	4, 8, 13	3eqtrd 2808	1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = {𝑧 ∈ dom 𝑋 ∣ (𝑋‘𝑧)𝑅𝐴})

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {cab 2747  {crab 3423   class class class wbr 5110  ^◡ccnv 5658  dom cdm 5659   “ cima 5662  Fun wfun 6527   Fn wfn 6528  ‘cfv 6533  (class class class)co 7408  ∘_RV/𝑐corvc 34787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-fv 6541  df-ov 7411  df-oprab 7412  df-mpo 7413  df-orvc 34788

This theorem is referenced by:  elorvc  34791